REVIEW
Resonance Energy Transfer: From Fundamental Theory to Recent 1
Applications 2
Garth A. Jones1*, David S. Bradshaw1* 3
1School of Chemistry, University of East Anglia, Norwich NR4 7TJ, UK 4
*Correspondence: 5
Corresponding Authors 6
[email protected], [email protected] 7
Keywords: Förster theory, FRET, electronic energy transfer, photosynthesis, solar harvesting, 8 plasmonics, cavity QED, interatomic Coulombic decay 9
10
Abstract 11
Resonance energy transfer (RET), the transport of electronic energy from one atom or molecule to 12
another, has significant importance to a number of diverse areas of science. Since the pioneering 13
experiments on RET by Cario and Franck in 1922, the theoretical understanding of the process has 14
been continually refined. This review presents a historical account of the post-Förster outlook on 15
RET, based on quantum electrodynamics, up to the present-day viewpoint. It is through this quantum 16
framework that the short-range, R–6 distance dependence of Förster theory was unified with the 17
long-range, radiative transfer governed by the inverse-square law. Crucial to the theoretical 18
knowledge of RET is the electric dipole-electric dipole coupling tensor; we outline its mathematical 19
derivation with a view to explaining some key physical concepts of RET. The higher order 20
interactions that involve magnetic dipoles and electric quadrupoles are also discussed. To conclude, 21
a survey is provided on the latest research, which includes transfer between nanomaterials, 22
enhancement due to surface plasmons, possibilities outside the usual ultraviolet or visible range and 23
RET within a cavity. 24
25
1 Introduction and the early years of RET 26
Resonance energy transfer (RET, also known as fluorescence resonance energy transfer, FRET, or 27
electronic energy transfer, EET) is an optical process, in which the excess energy of an excited 28
molecule – usually called the donor – is transferred to an acceptor molecule [1-4]; as depicted 29
Jones and Bradshaw RET: Theory to Applications
2
This is a provisional file, not the final typeset article
schematically in Figure 1. Fundamentally, RET involves two types of elementary particles: electrons 30
and photons. In RET, all the electrons (including the dynamically active electrons) are bound to the 31
nuclei of the molecules, and typically reside in their valence molecular orbitals. As such, the 32
individual electrons do not migrate between molecules during the transfer process, since the 33
molecular orbitals (the wavefunctions) do not overlap, but instead move between individual 34
electronic states within the molecules. This is fundamentally different to the ultra-short-range Dexter 35
energy transfer, where electrons do in fact migrate between molecules via covalent chemical bonds 36
[5]. In RET, on relaxation of the electron to a lower energy electronic state in the donor, the excess 37
energy is transported to the acceptor in the form of the emitted virtual photon – this transfer is 38
facilitated by dipole-dipole couplings between the molecules. In fact, photons play two distinct roles 39
towards the process: one as the mediator of donor-acceptor transfer, and the other as an external 40
energy source that promotes donor valence electrons into an electronic excited state, via an 41
absorption process prior to RET. 42
43
In 1922, the pioneering work of Cario and Franck enabled the earliest observation of RET [6-8]. 44
Their spectroscopy experiment involved the illumination of a mixture of mercury and thallium 45
vapours at a wavelength absorbed only by the mercury; the fluorescence spectra that results show 46
frequencies lines that can only be due to thallium. In 1927, the Nobel laureate J. Perrin provided the 47
first theoretical explanation [9]: he recognized that energy could be transferred from an excited 48
molecule to a nearby-unexcited molecule via dipole interactions. Five years later, his son F. Perrin 49
developed a more accurate theory of RET [10] based on Kallman and London’s results [11]. 50
Extending the works of both Perrins, Förster developed an improved theoretical treatment of RET 51
[12-14]. Förster found that energy transfer, through dipole coupling between molecules, mostly 52
depends on two important quantities: spectral overlap and intermolecular distance. He discovered the 53
now famous R–6 distance-dependence law for the rate of resonance energy transfer in the short-range. 54
Much later, in 1965, this distance dependence predicted by Förster was verified [15]. This led to the 55
‘spectroscopic ruler’ by Stryer and Haugland [16,17], a useful technique to measure the proximity of 56
chromophores and conformational change in macromolecules using RET. The next section, which is 57
more technical than the rest of the article, details the history of RET based on quantum 58
electrodynamics (QED); it can be safely skipped by readers more interested in the current 59
understanding of RET. 60
61
Jones and Bradshaw RET: Theory to Applications
3
2 Historical role of quantum electrodynamics in RET 62
2.1 The success of QED 63
Quantum electrodynamics is a rigorous and accurate theory – which is completely verifiable by 64
experiment [18] – that describes the interaction of electromagnetic radiation with matter. This 65
quantum field approach differs to other theories in that the whole system is quantised, i.e. both matter 66
and radiation are treated quantum mechanically. QED provides additional physical insights 67
compared to classical and semi-classical electrodynamics, which treats electromagnetic radiation 68
only as a non-quantised wave. For example, the wave-particle duality of light is uniquely portrayed 69
within QED but not semi-classical theories. However, despite their deficiencies, classical and semi-70
classical theories can still be useful since, often, they are easier to implement analytically and more 71
economic computationally. 72
73
The first major QED publication is credited to Dirac who, in 1927, wrote a description of light 74
emission and absorption that incorporated both quantum theory and special relativity [19]; this 75
depiction later became known as the relativistic form of QED, which is used in systems that contain 76
fast moving electrons. Three years later Dirac completed his classic book ‘The Principles of 77
Quantum Mechanics’ [20] in which, among other exceptional works, he derived a relativistic 78
generalisation of the Schrödinger equation. However, for elementary physical quantities such as the 79
mass and charge of particles, calculations using this early form of QED produce diverging results. In 80
the late 1940s, this problem was resolved (by renormalisation) leading to a complete form of QED 81
developed independently by Feynman [21-25], Schwinger [26-29] and Tomonaga [30,31] – all three 82
procedures were unified by Dyson [32]. 83
84
The ability of QED to provide novel predictions is monumental, but its quantitative successes are 85
even more impressive. In particular, the theory accurately predicts the electronic g-factor of the free 86
electron to 12 decimal places. In Bohr magneton units, the most precise measurement of g/2 is 87
1.00115965218073(28) [33]; QED has a predicted value of 1.00115965218203(27) [34]. In addition, 88
there are other staggering quantitative successes. For example, the numerical calculation of Lamb 89
shift splitting of the 2S1/2 and 2P1/2 energy levels in molecular hydrogen predicts 1,057,838(6) kHz 90
[35], which is highly accurate compared to the experimental value of 1,057,839(12) kHz [36]. QED 91
also provides a number of predictions that are unobtainable by semi-classical theory. These include 92
Jones and Bradshaw RET: Theory to Applications
4
This is a provisional file, not the final typeset article
forecasts of spontaneous decay and the Casimir-Polder forces, a deviation from London forces for 93
long-range intermolecular interactions [37-41]. 94
95
2.2 Non-relativistic QED: a theoretical framework for RET 96
An individual RET process, which arises after excitation of the donor, involves light emission at one 97
molecule and light absorption at the other. Such light-molecule interactions are best described by 98
QED. This means that the quantum properties and the retardation effects of the mediating light, 99
which leads to the concept of a photon, is directly incorporated into the calculations. Therefore, in 100
terms of this framework, it is natural to describe RET in terms of photon creation and annihilation 101
events. Namely, the creation of a photon at the excited donor and a photon annihilation at the 102
unexcited acceptor. Mathematically, these couplings are represented as off-diagonal matrix elements 103
of the interaction Hamiltonian. A full quantum description is usually necessary to describe the RET 104
process over all distances, this is because the electronic energy is not transferred instantaneously as 105
assumed by the classical and semi-classical descriptions (although retardation effects are sometimes 106
provided in such frameworks [42]). The transfer of energy between molecules occurs via the 107
exchange of a virtual photon, which has increasingly real (transverse) characteristics as the 108
intermolecular separation grows; this is discussed, in more detail, in Section 3.2. The term virtual 109
being indicative of the fact that the photon is reabsorbed before its properties, such as wavelength, 110
take on physical significance. The dipole of each molecule is also correctly described as a transition 111
dipole moment, connecting two non-degenerate energy states of the molecule. 112
113
Since RET involves slow moving electrons, bound within the valence states of the molecules, the 114
non-relativistic variant of QED (as opposed to relativistic or Lorenz gauge QED) is used. The theory 115
that underpins the quantum description of RET is the Power-Zienau-Woolley formalism of molecular 116
(or non-relativistic) QED [43-48], which utilises the Coulomb gauge, , where A��
is the 117
vector potential and the fields of the mediating photons can be naturally deconstructed into 118
longitudinal and transverse components. The longitudinal components, with respect to the 119
displacement vector R��
, are associated with the scalar potential and have a particular affinity for 120
coupling molecular transition moments in the near-zone, where the donor-acceptor pair are close 121
together. In regions far from the source (i.e. distant from the donor) the wave-vector k�
and R��
are 122
essentially collinear and the scalar potential approaches zero. In this case, the transverse part of the 123
Jones and Bradshaw RET: Theory to Applications
5
field dominates the coupling of the transition dipole moments of individual molecules [49]. This has 124
important implications for the spatial and temporal dynamics of excitons within molecular aggregates 125
[50,51]; namely, transition dipole moment pairs that are collinear to each other and collinear to the 126
displacement vector are coupled by the longitudinal components of the field only. 127
128
The QED model of RET is traceable to the 1966 paper by Avery, which extended the Perrin and 129
Förster theory of RET by replacing the Coulomb interaction with the relativistic Breit interaction 130
[52]. Although Avery did not explicitly include the effects of the mediating photon, in terms of the 131
creation and annihilation field operators, he nevertheless made a direct connection between RET and 132
spontaneous emission. Moreover, he determined the R–2 dependence on the transfer rate in the 133
far-zone. He concluded that investigating RET from the point-of-view of the ‘direct action’ 134
formulation of QED, devised by Wheeler and Feynman [53], would be ‘extremely interesting’. Soon 135
afterwards, in the same year, the Avery work was enhanced by a more formal and rigorous quantum 136
theoretical outlook provided by Gomberoff and Power [54]. 137
138
2.3 RET coupling tensor: the quest for its correct form 139
In the early 1980s there were a number of RET studies by Thirunamachandran, in collaboration with 140
Power and Craig, which give valuable insights into the physical connections between the near- and 141
far-zone mechanisms of RET. In 1983, Power and Thirunamachandran published three seminal 142
papers on QED theory [55-57]. Here they consider the problem within the Heisenberg formalism, 143
via the time evolution of operators associated with both electron fields and Maxwell fields. In the 144
third paper of the series, they derive an expression for the time dependent evolution of the RET 145
quantum amplitude as; 146
147
( ) ( ) ( )( )
( )( )
( ) ( )( )
( )( )
0 0 21
1 e 1 e 1sin
A D A
p qfi i j ij i j
ict k k ict k k
A D D A D
c t D Ac
kR dkR k k k k k k k k
µ µ δ
π
+∞ − −
−∞
= −∇ + ∇ ∇
− −× + − − − −
ℏ , (2.1)
148
149
where is the transition dipole moment of molecule X along the kth canonical coordinate and R 150
is the distance between the two molecules. The transfer occurs from an excited molecule D to 151
molecule A, initially in its ground state. Subscripts i and j represent Cartesian components with the 152
µk X( )
Jones and Bradshaw RET: Theory to Applications
6
This is a provisional file, not the final typeset article
usual tensor summation convention being employed [58]. The transition dipole moments elements 153
are ( )0µ pi D and ( )0µq
j A ; where molecule D is initially in state p, and the final state of molecule A is 154
q. Integration is over all possible wave-vectors (denoted by k) of the mediating photon. In this work, 155
the rapidly oscillating terms were dropped, to leave only two terms instead of the usual four; vide 156
infra, equation (2.6). The terms kD and kA represent the wave-vectors resonant with a transition of 157
molecules D and A, respectively. Power and Thirunamachandran did not explicitly describe how the 158
singularities in (2.1) were dealt with mathematically, but they show that the final expression 159
conforms to the correct distance dependencies in the appropriate limits. 160
161
Around the same time, Thirunamachandran and Craig considered resonance coupling between 162
molecules ‘where one was in an excited state’, within the dipole approximation (the term ‘resonance 163
energy transfer’ was not used in this work). They initially published the work as an extended paper 164
[59], and expanded upon it in their widely known book [45]. They consider two identical molecules 165
and calculate the interaction of the excited system D with the unexcited system A. Firstly, they 166
considered calculations that ignored retardation effects and any time explicit dependencies. The 167
calculated electric field at A, produced by the oscillating dipole at D, produces an energy change of; 168
169
( ) ( ) ( ) ( )1 3 0 00
ˆ ˆ4 3πε µ µ δ− −∆ = −p qi j ij i jE R D A R R . (2.2) 170
171
The final term is an orientational factor that modulates the magnitude of the energy difference based 172
on the relative dipole orientations of the molecules. Through the inclusion of retardation effects, 173
equation (2.2) becomes; 174
175
( ) ( ) ( )
( ) ( )
1 0 00
2 13 2
4 e
cos cos sinˆ ˆ ˆ ˆ3
p q ik Ri j
ij i j ij i j
E D A
kR kR k kRk R R R R R
R R R
πε µ µ
δ δ
− ⋅
−
∆ =
× − − + −
� �
.
(2.3) 176
177
Retardation effects give rise to the appearance of a phase factor, e ⋅� �
ik R , as well as two other distance 178
dependencies, namely, and . 179 R−1 R−2
Jones and Bradshaw RET: Theory to Applications
7
The authors then calculated the fully retarded matrix element in tensor-form and show that it is the 180
same as expression (2.3). The calculation formally involves summing over all photon wave-vectors 181
connecting the initial and final states. In practice, this summation involves using a box quantization 182
technique to transform the problem to an integral in momentum space. The solution can be found by 183
contour integration, in a way analogous to that in which Green’s functions solutions are found in 184
quantum scattering problems [60]. For identical molecules, the final matrix element (or quantum 185
amplitude) in tensorial form is: 186
187
( ) ( )0 0( , )µ µ=��
n mfi i ij jM D V k R A , 188
189
where; 190
191
. (2.4) 192
193
In light of the subsequent analysis shown later, it is important to note that the interaction tensor Vij , 194
derived in this early work, is purely the real part of the full expression. In deriving equation (2.4), 195
four different contours could be chosen around the two poles (the singularities), leading to different 196
results. The contour they chose ensures a correct outgoing-wave solution, although there is no a 197
priori mathematical basis for this choice. 198
199
Further advances were achieved by Andrews and co-workers who proved a direct relationship 200
between radiationless and radiative RET [61-63]. Although all three regimes of RET – i.e. the R–2, 201
R–4 and R–6 dependencies on the rate – were mathematically predicted in the original derivations, 202
Andrews et al. were the first to comment upon the relevance of the intermediate-zone contribution, 203
which has a R–4 dependence. This term dominates at critical distances; that is, when the distance 204
separating the molecules is in the order of the reduced wavelength, 2λ π=Ż , of the mediating 205
photon (i.e. R ~ Ż ). Inclusion of all three distance-dependencies in one rate equation is known as the 206
unified theory of RET. The particulars of which are provided in Section 3.2. 207
Jones and Bradshaw RET: Theory to Applications
8
This is a provisional file, not the final typeset article
Initially Andrews and Sherborne in 1987, reconsidered the problem in the Schrödinger 208
representation, where they derived the electric dipole-electric dipole tensor without the need of 209
‘outgoing wave’ arguments of scattering theory [59]. Starting from the second-order expression for 210
the time-dependent probability amplitude for energy transfer, they inserted all intermediate states to 211
obtain a rather complicated looking expression (not reproduced here). As detailed in the original 212
paper, the integral of the expression gives rise to four different Green’s functions, and hence four 213
choices of contour. The fact that four terms arise is attributed to the forward and reverse transfer 214
processes. They showed that the choice of contour was not unique, with each giving different 215
expressions for . Interestingly, they found that these new contours introduced imaginary 216
terms into , i.e. those not included in the derivations of the earlier work by 217
Thirunamachandran and Craig. By choosing the contour that appeared to be the ‘most acceptable’, 218
they derived the coupling matrix element to be of the form (corrected later by Daniels et al. [63] and 219
modifying the indexing here for better comparison with the expressions above): 220
221
, (2.5) 222
223
where, 224
225
, 226
, 227
228
in which σij is the expression given in (2.4). This derivation eliminates the need for physical 229
arguments based on quantum scattering theory used in the earlier work. It, nevertheless, did require 230
careful consideration of the correct contour with which to apply Cauchy’s residue theorem for 231
solving the integral. In later work, Andrews and Juzeliūnas applied an alternative method of contour 232
integration, whereby they infinitesimally displaced the problematic poles away from the real axis 233
[64]. The idea being that the imaginary addenda shifted the poles to enable integration around a 234
closed contour along the real axis. The approach gave results in agreement with those of Andrews 235
and Sherborne’s favoured choice of contour. Thus, this study removes the need to choose a contour; 236
Jones and Bradshaw RET: Theory to Applications
9
however, artificial displacements of the poles, including the choice of direction of displacement on 237
the complex plane, must be made. 238
239
In 2003, Daniels et al. re-examined the problem and avoided the uncertainties of the contour 240
integration entirely by solving the Green’s function using judicious substitutions within the integrals. 241
Namely, when the Green’s function is expressed as a sum of two integrals, so that; 242
243
( ) ( ) ( )0 0
sin sin,
pR pRG k R dp dp
R k p R k p
∞ ∞
= +− − − , (2.6) 244
245
substitutions of the form and for the first and second integral, respectively, 246
give an expression in which terms are oscillatory, but convergent. The authors solved these integrals 247
by expressing them as series expansions (in the form of special functions) to get a result, analogous 248
to equation (2.5), in the form: 249
250
251
. (2.7) 252
253
Here, on comparing with the earlier expressions, the only difference is a choice of sign for the 254
imaginary term τij. The authors suggested that the ambiguity of sign for this term signifies that 255
describes both incoming and outgoing waves, accommodating thereby both time-ordered 256
(Feynman) diagrams, as a complete quantum description should. However, the authors stress that it 257
is unimportant which sign to ascribe to a particular process (photon absorption or emission), as only 258
the modulus squared of the matrix element is physically measureable and, hence, using either sign on 259
τij provides an identical result for all calculations relevant to experiment. Jenkins et al. wrote a 260
follow-up paper that analysed the importance of each Feynman diagram, called time-ordered 261
pathways, to the overall RET rate. They discovered that both pathways have equal contribution 262
when the two molecules are close together; however, one pathway begins to dominate as the 263
molecules are moved further apart [65]. 264 265
t = pR− kR s= pR+ kR
Jones and Bradshaw RET: Theory to Applications
10
This is a provisional file, not the final typeset article
In 2016, Grinter and Jones re-derived expression (2.7) using a spherical wave description of the 266
mediating photon, via vector spherical harmonics [66]. All previous derivations employed a plane-267
wave description of the mediating photon. One advantage of the spherical wave approach is that 268
multipole contributions are more concretely defined in terms of the angular momentum quantum 269
numbers l and m. Furthermore, the work involved the development of an approach complementary 270
to the plane wave methods, giving additional insight into orientational aspects of RET and forming a 271
natural setting for the decomposition of fields into transverse and longitudinal components. In 2018, 272
a comprehensive review of the spherical wave approach was published [67]. In the plane-wave 273
method, defined in terms of the position vector r�
, the oscillating part of the field is expanded as; 274
275
( ) ( )2 3
1 1 12 3
e ...! !
ik r
n n
ik r ik re e ik r⋅
⋅ ⋅ = + ⋅ + + +
� �
� �� ��� � �
. (2.8) 276
277
where the first term relates to the electric dipole, the second to the magnetic dipole and the electric 278
quadrupole, and so on. In the spherical wave description, the expansion is written as; 279
280
( ) ( ) ( )e 2 1 cosik r l
l l
l
i l j kr P ϑ⋅ = +� �
. (2.9)
281
282
where are Bessel functions and are Legendre polynomials. The spherical wave 283
description consequently attributes radiation emerging from specific pure multipole sources to 284
specific angular momentum quantum numbers, thereby separating different multipole contributions 285
that are of the same order. 286
287
Additionally, derivation of the RET matrix element using spherical waves eliminates the need to 288
perform contour integration and, therefore, select the physically correct solutions. The arbitrary 289
choice of sign, which can be seen in the imaginary part (τij) of equation (2.7), does not appear in the 290
spherical wave analysis. The R dependence can be expressed in terms of Hankel functions of the first 291
kind, i.e. hl1( ) kR( ) = j l kR( ) + inl kR( ) for outgoing waves, while Hankel functions of the second kind, 292
i.e. hl2( ) kR( ) = j l kR( ) − inl kR( ) describe incoming waves. The ambiguous sign in equation (2.7) was 293
interpreted to mean that both incoming and outgoing waves are required to calculate the quantum 294
j l kr( ) Pl cosϑ( )
Jones and Bradshaw RET: Theory to Applications
11
amplitude of the process (i.e. photon absorption and emission). In the spherical wave approach, the 295
incoming and outgoing waves emerge naturally and can be linked directly to one or other of the signs 296
in the imaginary part of equation (2.7), up to the phase factor exp ±iωt( ) . 297 298
In a separate study, Grinter and Jones also analysed the transfer of angular momentum between 299
multipoles using a spherical description of the mediating photon [68]. Although it has been known 300
for some time that coupling between multipoles of different order can be non-zero [69-74], this work 301
showed that RET between multipoles of different order is formally allowed. This is because the 302
isotropy of space is broken during an individual transfer event, even though one may expect the 303
process to be forbidden on the grounds of the violation of the conservation of angular momentum. 304
For example, in the case of electric dipole-electric quadruple (E1-E2) transfer, two units of angular 305
momentum are lost from the electronic state of a quadrupole emitter (the donor), whereas the dipole 306
acceptor only takes up one quantum of electronic angular momentum. The above analyses indicate 307
that treating the mediating photon of an RET process in terms of spherical waves may be valuable in 308
some applications, particularly in the case of multipolar QED. A discussion on higher-order 309
considerations, such as these, is found in Section 3.3 310
311
3 RET based on quantum electrodynamics 312
3.1 Derivation of the RET coupling tensor 313
In order to understand any optical process within the framework of QED, a matrix element (or 314
quantum amplitude) that links the initial and final states is required. In the case of RET between two 315
molecules, the initial state is the donor, D, in an excited state and an acceptor, A, in the ground state. 316
In the final state, the acceptor molecule is in an excited state and the donor molecule is in its ground 317
state. Photophysically, this can be simply understood as; 318
319
D A D A∗ ∗+ → + , (3.1) 320
321
where, in this type of chemical expression, the asterisk denotes the molecule in an electronically 322
excited state. 323
324
Jones and Bradshaw RET: Theory to Applications
12
This is a provisional file, not the final typeset article
The usual starting point for any QED analysis is the illustration of the process by Feynman diagrams 325
[23], thereby aiding construction of the matrix element by defining all of the intermediate system 326
states. Feynman diagrams are graphical descriptions of electronic and photonic processes with a time 327
frame that moves upwards. Resonance energy transfer between two molecules, in isolation, involves 328
two Feynman diagrams – as shown in Figure 2. Here, examining the left-hand diagram, the initial 329
system state has the donor in excited state n and the acceptor in the ground state, labelled 0 (the red 330
section). Moving up the time axis, a photon is created from the excited donor to provide an 331
intermediate system state, in which both molecules are in the ground state and a photon is present 332
(the black section). Higher up the diagram this photon is annihilated at the donor and, thus, excites it 333
to state m (the blue section). The diagram on the right-hand side is legitimate, albeit counter-334
intuitive. In this case, the intermediate system state represents both molecules simultaneously in their 335
excited states in the presence of the mediating photon – meaning that conservation of energy is 336
clearly violated. However, this is fully justifiable within the constraints of the energy-time 337
uncertainty principles. 338
339
These diagrams (which represent the two pathways of RET) involve two light-molecule interactions: 340
one at the donor and the other at the acceptor. This is indicative of second-order perturbation theory, 341
which we examine below, as the minimal level of theory necessary to describe RET. The total 342
Hamiltonian for RET between neutral molecules, in multipolar form, is written as; 343
344
. (3.2) 345
346
Here, the first two terms correspond to the molecular Hamiltonians of the donor and acceptor347
, which are usually the non-relativistic Born-Oppenheimer molecular 348
Hamiltonian. The third term is the radiation Hamiltonian, , not seen in semi-classical theory; 349
this is typically defined in terms of the electric and magnetic field operators and/or the auxiliary field 350
operator, ( ),��
a R t [45,75]. Although these three Hamiltonians are important for describing the light-351
matter system in its entirety, they play no explicit role in the derivation of the matrix element for 352
RET. The key parts of the Hamiltonian for RET are the interaction terms . These 353
two terms represent the interaction between each molecule and the electromagnetic field; they are 354
perturbative in nature because the light-molecule interactions of RET is weak compared to the large 355
H = Hmol D( ) + Hmol A( ) + H rad + H int D( ) + H int A( )
Hmol X( ); X = D, A
H rad
H int X( ); X = D, A
Jones and Bradshaw RET: Theory to Applications
13
Columbic energies of the molecules. The eigenstates of the interaction Hamiltonian are constructed 356
with the tensor product of molecule and radiation states. Of particular note is that no interaction term 357
between the donor and acceptor exists in equation (3.2), unlike in semi-classical formalisms. The 358
QED description of RET is, therefore, a genuinely full quantum theory, whereby the transfer of 359
energy between an excited donor to an unexcited acceptor is via the electromagnetic field; direct 360
Coulombic interactions between the two molecules do not arise in this multipolar form of the 361
Hamiltonian [55]. 362
363
Using the electric dipole approximation, in which only the transition electric dipole (E1) of each 364
molecule are considered, the interaction Hamiltonian is written as; 365
366
( ) ( ) ( ) ( )1 1
int 0 0ε µ ε µ− ⊥ − ⊥= − ⋅ − ⋅�� ��� �� �
D AH D d R A d R , (3.3) 367
368
where ( )Xµ��
is the dipole operator of molecule X at position XR��
(it is usually presumed that the donor 369
is positioned at the origin); ε0 is the permittivity of free space. The displacement electric field 370
operator, ( )Xd R⊥���
, can be written in terms of a mode expansion; 371 372
( ) ( ) ( ) ( ) ( ){ }1
2( ) ( ) ( ) †( )0
,
e e2
λ λ λ λ
λ
ε⊥ ⋅ ∗ − ⋅ = −
�� ��� �
�
��� ℏ � � � � � �X Xip R ip RX
p
cpd R i e p a p e p a p
V . (3.4) 373
374
Here, c is the speed of light in a vacuum, ( )( )e pλ� � defines the polarisation of the mediating photon 375
(the asterisk denoting its complex conjugate), ( )( )a pλ � and ( )†( )a pλ �
are the annihilation and creation 376
operators, respectively, for a photon of wave-vector p�
and polarisation . In the pre-exponential 377
factor, represents the volume used in the box quantisation procedure that enables fields to be 378
defined in terms of operators, as required by QED. The second-order perturbative term, which is the 379
leading term in the matrix element for RET, is explicitly written (in terms of Dirac brackets) as; 380 381
1 2
int 1 1 int int 2 2 int= +− −fi
i I i I
f H I I H i f H I I H iM
E E E E . (3.5) 382
383
λV
Jones and Bradshaw RET: Theory to Applications
14
This is a provisional file, not the final typeset article
From Figure 2, we easily identify the key system states (which is a combination of the two molecular 384
states and the radiation state). These are the initial state ( )0, ;0 ,nD Ai E E p λ= �
(donor excited, 385
acceptor unexcited and no photon), the final state ( )0 , ;0 ,mD Af E E p λ= �
(donor unexcited, acceptor 386
excited and no photon) and the two possible intermediate states, ( )0 01 , ;1 ,D AI E E p λ= �
(donor and 387
acceptor unexcited and one photon) and ( )2 , ;1 ,n mD AI E E p λ= �
(donor and acceptor excited and one 388
photon). The radiation states, often referred to as number or Fock states, have eigenvalues that are 389
occupation numbers of the quantized electromagnetic field, i.e. the number of photons in the system. 390
The creation and annihilation operators act on the relevant radiation states via 391 †( )( ) 0( , ) 11( , )λ λ λ=� � �
a p p p and ( )( ) 1( , ) 10( , )λ λ λ=� � �a p p p . The commutator involving these two 392
operators is given by the relationship ( ) ( )1( ) †( ) 3 3( ), ( ) 8λ λλλπ δ δ
−′′′ ′ = −
� � � �a p a p V p p , where ( )δ ′−� �p p 393
is a Dirac delta function and λλδ ′ is a Kronecker delta [76]. 394
395
Equipped with these state expressions, the interaction Hamiltonian of equation (3.3) and the energies 396
of each state in Table 1 (note that the initial and final states have the same energy, since conservation 397
of energy has to be restored after a miniscule amount of time), an expression for the RET matrix 398
element can be found. For illustrative purposes, we explicitly calculate just one of the Dirac 399
brackets, namely ; which is the initial bracket, since it is convention to move from right to 400
left in these equations. Explicitly, it is written as; 401
402
( ) ( ) ( ) ( ) ( ) ( )0 0 1 1 01 int 0 0, ;1 , , ;0 ,λ ε µ ε µ λ− ⊥ − ⊥= − ⋅ − ⋅
�� ���� � � �nD AD A D AI H i E E p D d R A d R E E p . (3.6) 403
404
This represents the creation of a photon when the excited donor relaxes (the acceptor is unchanged, 405
as denoted by the superscript on either EA) and, hence, dipole operators acting on the acceptor 406
molecular state and the annihilation operator (within d⊥) on the radiation state are zero due to 407
orthonormality. Therefore, equation (3.6) is simplified to; 408
409
( ) ( ) ( ) ( )1 01 int 0 1 , 0 ,ε µ λ λ− ⊥= −
���� � �nDD DI H i E D E p d R p . (3.7) 410
411
The solution of which, on insertion of equation (3.4), is expressed concisely as; 412
413
I1 H int i
Jones and Bradshaw RET: Theory to Applications
15
( ) ( )( )
1
200
1 int,
e2
λ
λ
ε µ∗ − ⋅ =
���
�
ℏ � Dn ip Ri i
p
cpI H i i e p D
V , (3.8) 414
415
with the i th component of the transition dipole moment written as; 416
417
( ) ( )0 0n n
i D i DD E D Eµ µ= . (3.9) 418
419
Following a similar procedure for the other three Dirac brackets, and finding the energy 420
denominators for each term of (3.5), the full expression for the RET process is given as; 421
422
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )0 0 0 0
, 0 0 0
e e
2
ip R ip Rn m n m
fi i j i j j ip n n
cpM e p e p D A D A
V E cp E cpλ λ
λµ µ µ µ
ε
⋅ − ⋅∗ = + − − −
� �� �
�
ℏ � �
ℏ ℏ . 423
(3.10) 424
425
In order to determine a final result for the RET matrix element, we use the cosine rule to rewrite the 426
summation over of polarizations as; 427
428
( ) ( ) ( ) ( ) ˆ ˆλ λ
λδ∗ = −
� �i j ij i je p e p p p , (3.11) 429
430
where δij is the Kronecker delta and a caret denotes a unit vector, and convert the inverse of the 431
quantization volume to an integral in momentum space; 432
433
( )3
3
1
2p
d p
V π→
�
�
. (3.12) 434
435
The quantum amplitude then becomes an integral of the form; 436
437
( ) ( ) ( )
( ) ( ){ } ( )
0 02 2
0
3
3
1ˆ ˆ
2
e e e +e ,2
µ µ δε
π⋅ − ⋅ ⋅ − ⋅
= −−
× − +
� � � �� � � �
�
n mfi i j ij i j
ip R ip R ip R ip R
pM D A p p
k p
d pk p
(3.13)
438
Jones and Bradshaw RET: Theory to Applications
16
This is a provisional file, not the final typeset article
where ħck is the energy transferred from D to A. As outlined in the subsequent section, this integral 439
has been solved analytically using various vector calculus techniques. Omitting the long and 440
intricate derivation based on special functions [63], the matrix element for RET – including the 441
retarded electric dipole-electric dipole (E1-E1) coupling tensor, denoted as Vij – is obtained as; 442
443
( ) ( )0 0( , )µ µ=��
n mfi i ij jM D V k R A , (3.14) 444
( ) ( ) ( ) ( ){ }2
30
e ˆ ˆ ˆ ˆ( , ) 1 34
ikR
ij ij i j ij i jV k R ikR R R kR R RR
δ δπε
= − − − −��
. (3.15) 445
446
A more in-depth analysis of the derivation of the E1-E1 coupling tensor, Vij, and the transfer rate of 447
RET (an outline of which follows) – without providing all of the intricate specifics – is delivered by 448
Salam in his recent review [77]. 449
450
3.2 Physical interpretation of the RET coupling tensor 451
The physical observable derived from the Vij tensor, via the matrix element, is the transfer rate of 452
RET, symbolised by Γ. This rate is demined from the Fermi rule [78]: 2
2π ρΓ = ℏ fi fM , where ρf 453
is the density of acceptor final states. Assuming a system of two freely tumbling molecules, meaning 454
that a rotational average is applied [79], the following is found; 455 456
( ) ( ) ( )2 21~ A ,
9µ µΓ � �
D A k R . (3.16) 457
458
where the E1-E1 transfer function, ( )A ,k R , is defined by [62]; 459
460
( )( )
( ) ( ){ }2 4
230
2A , ( , ) ( , ) 3
4πε∗= = + +
�� ��
ij ijk R V k R V k R kR kRR
. (3.17) 461
462
In contrast to Förster coupling, the QED form of the electronic coupling has a complicated distance 463
dependence, which underscores the unification of the radiationless and radiative transfer 464
mechanisms. Whereas the semi-classical Förster theory predicts only an R–6 dependence [80], the 465
Jones and Bradshaw RET: Theory to Applications
17
QED rate expression of (3.17) contains three distance dependencies: R–2, R–4 and R–6. This signifies 466
three distinct regimes that dominate in the long-, intermediate- and short-range, respectively. 467
468
The different regimes of RET are most readily understood in terms of the mediating photon [49]. As 469
outlined in Section 2.2, the photon is said to have real characteristics – i.e. it has a large transverse 470
component w.r.t. �R – when the separation of the donor and acceptor exceeds its reduced wavelength 471
(i.e. ≫ŻR ). Meaning that, since the mediating photon is always transverse w.r.t. its wave-vector 472
p�
, the photons (emitted in all directions by D) that are annihilated at A in the long-range are the ones 473
where p�
is essentially co-linear with �R . Conversely, if R is significantly less than the reduced 474
wavelength the photon is fully virtual, meaning that retardation effects are not present. That is, it 475
does not have well defined physical characteristics, such as momentum. This arises because, due to 476
the uncertainty principle, the position of the mediating photon is ‘smeared out’ in the short-range so 477
that p�
may no longer be co-linear with �R – therefore, there is a longitudinal component to the 478
photon w.r.t. �R . The two limiting cases of RET are, hence, often referred to as radiationless (virtual 479
photon) and radiative (real) transfer – in the past, until the unified theory, they were usually 480
considered to be two completely separate and distinct mechanisms. Since all three terms of equation 481
(3.17) are non-zero in RET (or, at least, the short-range term always exists), it is justifiable to say that 482
all photons are virtual in nature [49,81]. This means that a notional ‘real’ photon – which is 483
transverse w.r.t both p�
and �R – does not exist, because these two vectors are never exactly collinear 484
due to the uncertainty principle. 485 486
To summarise, long-range (or far-zone) energy transfer has an inverse-square, R–2, dependence on the 487
rate, and short-range (or near-zone or Förster) transfer has the well-known R–6 dependence. That 488
leaves the intermediate zone, which was not previously identified until Andrews’s work [62], where 489
the distance separating the molecules is of the same order as the reduced wavelength of the mediating 490
photon; this region has an R–4 dependence. Our expressions have assumed dynamic coupling 491
between the transition dipole moments of the donor and acceptor, for cases of static dipole couplings 492
(in which k = 0) only the first term of equation (3.17) applies. 493
494
495
496
Jones and Bradshaw RET: Theory to Applications
18
This is a provisional file, not the final typeset article
3.3 Higher order RET 497
Often the electric dipole approximation is employed for studies on RET, which means that only E1-498
E1 coupling is considered. However, the coupling of the electric dipole of a molecule with the 499
magnetic dipole (M1) or electric quadrupole (E2) of the other can be important [82], for example, in 500
chirality-sensitive RET [77,83-86]. E1-M1 and E1-E2 couplings are, in general, of similar 501
magnitude but are roughly 150 times smaller than E1-E1 interactions; other multipoles are even 502
smaller and almost never utilised in RET analyses. 503
504 The derivation of the matrix element for E1-M1 coupling, with use of special functions, is provided 505
elsewhere [63]. The final result is given by; 506
507 508
( ) ( ) ( ) ( ) ( )0 0
E1-M1 0 0 ,n mj jm n
fi i i ij
m A m DM D A U k R
c cµ µ = +
� , (3.18) 509
510
which features the transition magnetic dipole, mj, and the E1-M1 tensor, ( ),ijU k R�
, with the latter 511
explicitly expressed as; 512
513
( ) ( )2 23
0
ˆe,
4
ikRk
ij ijk
RU k R ikR k R
Rε
πε
−
= − +�
, (3.19) 514
515
where εijk is the Levi-Civita symbol. Following a rotational average [79], the rate of RET based on 516
this type of coupling is; 517
518
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){ }2 2 2 2
2
B ,2Re .
9µ µ µ µ∗ ∗′Γ + − ⋅ ⋅� � � � � � � �
∼k R
D m A A m D D m D A m Ac
(3.20) 519
520
where the E1-M1 transfer function, ( )B ,k R , is written as; 521
522
( ) ( ) ( ) ( )( )2 2 4 4
230
2B , , ,
4πε∗= = +
� �
ij ijk R U k R U k R k R k RR
. (3.21) 523
524
Jones and Bradshaw RET: Theory to Applications
19
Comparing equations (3.17) with (3.21), i.e. the A and B functions, it is clear that the first term (the 525
R–6 dependent term) is missing in E1-M1 coupling. Physically, this means that the photons that 526
mediate E1-M1 interactions have real characteristics, i.e. they are never fully virtual. However, in 527
contrast to a commonly held view, E1-M1 coupling is not exclusively related to radiative energy 528
transfer since a short-range R–4 term also exists. The lack of the R–6 term also tells us that static 529
electric and magnetic dipoles (in which k = 0) do not interact, since all the other terms involve k. 530
531
The matrix element for E1-E2 interactions is determined as [69,71]; 532
533
( ) ( ) ( ) ( ){ } ( )E1-E2 ( , )µ µ ±= −
�
fi i jk jk i i jkM D Q A Q D A V k R , (3.22) 534
535
where the E1-E2 tensor, ( ) ( , )i jkV k R±�
, is expressed by; 536
537
( ) ( ){ ( )
( ) ( )
i2 2
40
2 2 3 3
e ˆ ˆ ˆ ˆ ˆ ˆ( , ) 3 3i 54
1 ˆ ˆ ˆ ˆ ˆi .2
kR
ij k jk i ki j i j ki jk
ij k ik j i j k
V k R kR k R R R R R R RR
k R k R R R R R R
δ δ δπε
δ δ
= − + + + + −
+ − + −
�
(3.23)
538
539
This expression is the -jk index symmetry form of the tensor, which is justified since it contracts with 540
the index-symmetric electric quadrupole, jkQ . After a rotational average, the corresponding rate is 541
obtained as; 542
543
( ) ( ) ( ) ( ) ( ) ( ) ( ){ }2 2C ,
15 λµ λµ λµ λµµ µ∗ ∗′′Γ +� �∼
k RD Q A Q A A Q D Q D , (3.24) 544
545
where ( )C ,k R is found as; 546
547
( ) ( ) ( ) ( ){ }2 2 4 4 6 6
240
1C , ( , ) ( , ) 90 18 3
4πε∗= = + + +
� �
i jk i jkk R V k R V k R k R k R k RR
. (3.25) 548
549
Jones and Bradshaw RET: Theory to Applications
20
This is a provisional file, not the final typeset article
Examining this expression, we see that E1-E2 coupling has four terms with the distance 550
dependencies R–2, R–4, R–6 and R–8 (rather than the three of E1-E1 interactions). The new 551
radiationless (R–8) term dominates in the near-zone, as predicted by Dexter [5], while the usual 552
inverse-square distance dependence of radiative transfer dictates the far-zone. The presence of these 553
terms (and the distinctive middle terms) in a single expression again signifies that they are the two 554
extremes of a unified theory. Since the first term does not depend on k, we determine that static 555
electric dipole and quadrupoles can interact. 556
557
3.4 Effects of a bridging molecule 558
Recent theoretical work, based on QED in the electric dipole approximation, is an analysis on the 559
effects of a third molecule, M, on RET [87-91]. In this sub-section, we touch upon the case where M 560
bridges the energy transfer between D and A – a Feynman diagram of which is provided in Figure 3. 561
This is the DMA configuration; the other cases (DAM and MDA), in which the molecules are 562
interchanged, have also been investigated. The matrix element for DMA, delivered from fourth-order 563
perturbation theory, is given by; 564
565
( ) ( ) ( )0 00 0( , ) ( , )DMA n mDM MAfi i ij jk kl lM D V k R M V k R Aµ α µ=�� ��
, (3.26) 566
567
where ( )00α jk M is the polarisability tensor that arises because two light-molecule interactions occur 568
at the third molecule (which begins and ends in its ground state, as denoted by the superscript 00) and 569
two couplings tensors are used since two energy transfer steps occur. Using the Fermi rule, the 570
leading term in the physically observable rate (that includes the third body) is the quantum 571
interference, i.e. the cross-term, that involves multiplication of equations (3.14) and (3.26) so that 572
[88]; 573
574
( ) ( ) ( ) ( ) ( )0 00 0 0 0( , ) ( , ) ( , )DMA DA n m n mDM MAfi fi i ij jk kl l p pq qM M D V k R M V k R A D V k R Aµ α µ µ µ∗ ∗=�� �� ��
. (3.27) 575
576
This is the rate that dominates if energy transfer between D and A is forbidden, for example, due to 577
symmetry selection rules or when the dipole moments of D and A are both orthogonal with each 578
other and their displacement vector, R��
. In this scenario, the mediator M facilitates the RET that 579
Jones and Bradshaw RET: Theory to Applications
21
would not occur otherwise [89]. A recent review by Salam provides a more comprehensive analysis 580
on the role of a third body in RET [77]. 581
582
4 Recent RET research 583
4.1 Nanomaterials for energy transfer 584
While the generic term ‘molecule’ has been used throughout this manuscript, other materials can be 585
used in RET such as atoms, chromophores, particles and, more recently, carbon nanotubes [92-96] 586
and quantum dots (QDs). In 1996, first observation of energy transfer between the latter was 587
achieved with cadmium selenide (CdSe) QDs [97] and similar compounds followed; for example, 588
cadmium telluride (CdTe) [98] and lead sulfide (PbS) [99] QDs. In experiments, quantum dots are 589
attractive because they can be much brighter, and contain greater photostability, than typical organic 590
chromophores [100,101]. Hence, QDs have become important in bio-inspired RET-based 591
applications [102,103], such as nanosensors [104-111] and photodynamic therapy [112,113]. In 592
terms of theory, it has been determined that RET between quantum dots and nanotubes can be 593
modelled using dipole-dipole couplings [90,114-119]. For more on the experiments and theory of 594
RET in nanomaterials, Liu and Qiu provide an excellent review on recent advances [120]. 595
596
While quantum dots are suggested as artificial antennas in synthetic light-harvesting materials 597
[111,121], research on such systems usually involve multi-chromophore macromolecules. One type 598
of which are known as dendrimers; from its periphery to core, these branch-like structures comprise 599
decreasing number of chromophores [122-130]. They work on the principle that photons are 600
absorbed at the periphery and the excitation energy is funnelled to a central reaction centre via 601
multiple RET steps; an example of this is shown in Figure 4. A significant amount of theory has 602
been published on this multi-chromophore transfer mechanism [131-140]. Towards the centre of the 603
dendrimer, where the number of chromophores is decreased, there is a possibility that two excited 604
donors will be in the vicinity of an acceptor. In this case, another RET mechanism, known as energy 605
pooling [141-143], becomes possible. This process is illustrated in Figure 5 and can be written, in 606
terms of photophysics, as; 607
608
D D A D D A∗ ∗ ∗∗+ + → + + , 609
610
Jones and Bradshaw RET: Theory to Applications
22
This is a provisional file, not the final typeset article
where the double asterisk denotes that the acceptor is doubly excited, i.e. the acceptor is promoted to 611
an excited state that requires the excitation energies of the sum of the two donors. This contrasts to 612
the process known as energy transfer up-conversion [144,145], which has the same initial condition 613
but excitation is transferred from one donor to the other – so that one of the donors is doubly excited 614
– and the third molecule is not involved. The matrix element for energy pooling has an analogous 615
form to equation (3.26); the only difference is that the superscript m0 on A (which is now a donor) 616
becomes 0n and the superscript 00 on M (now the acceptor) becomes s0, where s signifies a doubly 617
excited molecule. In recent years, Lusk and co-workers have demonstrated energy pooling 618
experimentally [146] and discovered, among other advances, that the efficiency of energy pooling 619
can be improved within a cavity [147-149]. Lately, moreover, they have studied the time-inverse 620
mechanism of energy pooling, known as quantum cutting, which involves the excitation on A 621
transferring to both D molecules [150]. 622
623
Another double-excitation mechanism is two-photon RET [151,152], which involves the absorption 624
of two photons at the donor and the transfer of the resulting excitation to the acceptor. The matrix 625
element of this process is identical to equation (3.14), except the superscript on D is 0s rather than 626
0n. Since the incident light in two-photon RET is lower in energy compared to RET, photo-627
destruction of living tissue can be circumvented. Therefore, biological applications of this process 628
have arisen, including photodynamic therapy [153-160] and bioimaging [155,160-163]. 629
630
4.2 Plasmon-based RET 631
The quest for control of light-energy at the nanoscale has led to some very interesting studies, from 632
both an experimental and a theoretical point-of-view, that often involve RET coupling between 633
molecules near a surface plasmon [164-194] – the latter, basically, acting as a bridging material for 634
the energy transfer. Plasmons are the collective excitations of conduction electrons by light, which 635
generally reside in a confined metallic structure. By coupling plasmonic materials to RET 636
chromophores, a substantial amount of energy transfer can occur over significantly larger separations 637
than the RET between conventional materials – up to distances approaching the optical wavelength. 638
The effects of a surrounding nanophotonic environment, such as a surface plasmon, on RET is an 639
ongoing debate [189,195]. 640
Jones and Bradshaw RET: Theory to Applications
23
In 2011, Pustovit and Shahbazyan developed a classical theory of plasmon-assisted RET that 641
involves an isotropic complex polarizability [196]. Their model, which maintains an energy balance 642
between transfer, dissipation and radiation, analyses the geometry of a plasmon-RET system – with a 643
focus on distance and orientational effects – by providing numerical results. This mechanism shows 644
that plasmon-assisted RET will dominate the usual non-radiative (Förster) transfer, even in the near-645
zone. While a comparable study predicts, over hundreds of nanometres, an enhanced rate by a factor 646
of 106 [197]. These forecast improvements now have experimental verification. For example, 647
Wenger and co-workers demonstrate enhanced transfer between donor-acceptor pairs confined to a 648
gold ‘nanoapparatus’; they endorse a six-fold increase in the rate of RET over 13 nm [198]. 649
In the years that followed, other innovative studies on plasmon RET have arisen. An experimental 650
study by Zhao et al. showed that the efficiency of RET can be controlled by the plasmonic 651
wavelength [199]. Remarkably, they discovered that RET can be turned off and on by tuning the 652
plasmon spectrum with the donor emission and acceptor absorption peaks, respectively. Related 653
theory develops the concept of a ‘generalised spectral overlap’, whereby the rate of plasmon RET is 654
not just dependent on the overlap integral of the donor emission and acceptor absorption spectra (as 655
follows from Förster theory), but includes a plasmonic contribution from an electromagnetic 656
coupling factor [200,201]. Other experimental work, which is analogous to the effects of a bridging 657
molecule that is discussed earlier [89], use plasmonic nanoantennas to enable E1-E1 RET that is 658
otherwise forbidden by geometry [202]. 659
Bershike et al. explain, by comparing model and experimental data, enhanced coupling between a 660
nanoscale metal and a light emitting dipole [203]. They employ a complex dielectric function that 661
indicates an R–4 distance dependence (ranging from 0.945 to 8.25 nm) between the fluorescent 662
molecule and the gold nanoparticle surface. Similar to this study, Bradley and co-workers provide an 663
investigation, which employs a Green’s tensor analysis of Mie theory, that again show plasmon RET 664
can display an R–4 dependence [204]. These results are consistent with numerical predictions, based 665
on QED, that intermediate-zone RET dominates at these separation distances [51]. 666
667
4.3 Energy transfer at non-optical frequencies 668
Resonance energy transfer usually occurs in the ultraviolet or visible range of the electromagnetic 669
spectrum, which is comparable to the energy required for electronic transitions in molecules etc. 670
Jones and Bradshaw RET: Theory to Applications
24
This is a provisional file, not the final typeset article
Recently, however, energy transfer involving either a much lower or higher frequency range has 671
gained traction. An outline on which now follows. 672
At the lower end, in the infrared range, transfer of vibrational energy can arise between excited 673
(donor) and unexcited (acceptor) oscillating bonds on adjacent molecules. Applications include the 674
observation of local orientational order in liquids [205] and, analogous to the spectroscopic ruler in 675
RET, a measure of intermolecular distances at the sub-nanoscale in the condensed phase [206,207]. 676
This type of transfer is especially prevalent between water molecules, due to the strong dipole-dipole 677
interactions between the O–H stretch vibrations [208-210]. It has been determined that, with some 678
modifications, that Förster theory can be valid at these light frequencies [211]. Energy transfer at 679
even lower frequencies, namely in the microwave range, is the subject of a very recent paper by 680
Wenger and co-workers [212]. In this work, the energy transfer is enhanced by positioning the donor 681
and acceptor pair within a cavity. 682
At the higher end is interatomic and intermolecular Coulombic decay (collectively ICD), a process 683
that involves the x-ray range of the spectrum. First predicted in 1997 [213], and experimentally 684
verified six years later [214], ICD is a process in which photoionization of one atom or molecule can 685
lead to remote photoionization of another atom or molecule via the exchange of a high energy 686
photon. In terms of fundamental theory, ICD is now understood to be equivalent to Förster transfer 687
(although ICD involves much more complex prior and posterior processes) – since the mechanism is 688
driven by dipole-dipole coupling with the characteristic R–6 distance dependence. Nevertheless, there 689
is a major fundamental difference between RET and ICD. Namely, as explained previously, the 690
former typically involves only valence electrons whereas ICD is initiated by an intra-atomic (or intra-691
molecular) decay process; a high-energy transition, in which a donor valence electron relaxes to the 692
core shell resulting in promotion of an acceptor valence electron to the continuum, i.e. acceptor 693
ionization. This means that an ionization cross-section will feature instead of the absorption cross-694
section of Förster transfer. 695
A prototypical example is the photo-ionization of a neon dimer (Ne2) via 2S-electron emission from 696
one of its atoms. This results in the relaxation of a valence 2P-electron into the formed vacancy and, 697
consequently, a high-energy photon is released. Following absorption of this light by the 698
neighbouring atom, a 2P-electron is ejected from it [215,216]. The interaction of the two newly 699
charged ions causes a Coulomb explosion, i.e. the fragmentation of the dimer. For clarity, the whole 700
mechanism is illustrated in Figure 6. ICD is typically ultra-short-range, in which (just like Dexter 701
Jones and Bradshaw RET: Theory to Applications
25
transfer) wavefunction overlap occurs; hence, terms relating to electron correlation and exchange will 702
contribute. Moreover, since ICD involves electron relaxation from a valence shell to the core shell in 703
the donor, account of the Auger effect is required. This competing mechanism occurs because the 704
energy generated from this relaxation could be transferred to another electron within the donor (and, 705
thus, ejecting it), so energy in the form of a photon would not reach the acceptor. Therefore, for an 706
accurate theoretical description of ICD, a detailed interpretation of the Auger effect along with 707
electron correlation and exchange is required. This is achieved by considering direct and exchange 708
Coulomb integrals for the decay rate. An overview of this is provided by Jahnke in his recent review 709
[217]. 710
Since the pioneering studies on diatomic systems, there have been a number of experimental and 711
theoretical investigations into ICD that involve different materials, including clusters of atoms and 712
molecules [218], quantum dots [219,220] and quantum wells [221]. Although ICD has considerable 713
theoretical interest, there is evidence of its practical importance to biological chemistry; in particular, 714
in the understanding of a DNA repair mechanism provided by the enzymes known as photolyases 715
[222,223]. The theoretical developments of ICD often mirror those already established in RET – 716
such as the effects of retardation, dielectric environments, a third body and virtual photons [224,225]. 717
Clearly, more research in this exciting emerging field is required, with much still to learn in terms of 718
its fundamental theory and applications. 719
720
4.4 RET in cavities 721
It can be challenging to elucidate fundamental processes experimentally, particularly because RET 722
often occurs in natural biological systems and ‘energy materials’ in the condensed phase. 723
Necessarily involving a level of phenomenological modelling, their simulation can be tremendously 724
complicated. Associated research, especially in connection to the field of biology, has been covered 725
in a numerous recent reviews [226-246]. Cavity quantum electrodynamics (cQED) works on the 726
principle that electronic species are restricted to small volumes (usually bounded by mirrors in one or 727
more dimensions) so that the electromagnetic field is tuned to specific quantised modes and the 728
quantum nature of the light becomes more apparent compared to the free field. In terms of 729
mathematical formulation, the arbitrary quantisation volume, V, of equation (3.10) is simply replaced 730
by the dimensions of the cavity. Early applications of cQED revealed an understanding of the 731
fundamental light-matter interactions in atoms, quantum dots and similar materials [247-252]. 732
Jones and Bradshaw RET: Theory to Applications
26
This is a provisional file, not the final typeset article
More recently cQED has been applied to chemical substances, such as organic dyes, and connected 733
to phenomena such as RET [253]. The main advantage of studying these cavity-based schemes is 734
that experimentalists are able to control the electromagnetic radiation at the quantum level, while 735
simultaneously reducing interference with the surroundings to a significant extent. This allows for 736
the explicit study of polariton modes (sometimes called hybrid states in this context), which is 737
typically difficult in the condensed phase because of the rapid decoherence that derives from system 738
coupling with a continuum of environmental modes. For example, in 2012, Ebbesen and co-workers 739
experimentally showed that the photophysical properties of light-induced chemical reactions can be 740
influenced by cavity fields, which can modify the chemical reaction landscape [254]. In another 741
study, the same research group cleverly showed how to alter the reaction rates of chemical reactions 742
by coupling molecular vibrations to infrared cavity modes [255]. 743
Since experiments with negligible amount of decoherence are now conceivable, there is increasing 744
interest in the effects of polariton modes on energy transfer within a cavity. In 2015, for instance, a 745
couple of theoretical studies indicated that ‘exciton conductance’ could be considerably enhanced, by 746
orders of magnitude, when organic materials are coupled to cavity modes [256,257]. Experimental 747
verification of this amplified energy transfer soon followed [258-260]. Attempts to better understand 748
polariton-assisted RET are increasingly prevalent. In 2018, Du et al. developed a ‘polariton-assisted 749
remote energy transfer’ model to explain how enhanced RET is mediated by vibrational relaxation in 750
an optical microcavity [261]. While earlier this year, Schäfer et al. proposed that energy transfer 751
could be drastically affected by a modification of the vacuum fluctuations in the cavity. In this 752
research, they make a connection to Förster and Dexter transfer, and account for the often-753
disregarded Coulomb and self-polarisation interactions. Interestingly, they predict that photonic 754
degrees of freedom give rise to electron-electron correlations over large distances in the cavity [262]. 755
What we do know for sure is that cavity RET is a representative example of the strong coupling 756
regime; an excellent recent review on such strong light-matter interactions is provided by Börjesson 757
and co-workers [263]. 758
759
5 Discussion 760
Today it is nearly 100 years since the discovery of RET and, remarkably, the 71 year-old Förster 761
theory that describes this transfer is still widely utilised. This model has provided us with the famous 762
R–6 distance dependence on the rate between donor and acceptor molecules. Following these earlier 763
Jones and Bradshaw RET: Theory to Applications
27
times, from the 1960s until the late 1980s, significant theoretical developments based on fundamental 764
quantum electrodynamics has been applied to two-centre RET. This has culminated into the unified 765
theory of RET, which links the short-range (near-zone) process of Förster with a long-range (far-766
zone), R–2 dependent transfer consistent with Coulomb’s Law. It also predicts a R–4 dependence in 767
the intermediate region, where the distance between the molecules approximately equals the reduced 768
wavelength of the mediating virtual photon. The latter could be said to have increasingly real 769
characteristics in this range. Although not detailed in this review, further work in the 1990s predicted 770
that optically active molecules in the condensed phase could also have a R–3 and a R–5 distance 771
dependence, which become significant when the imaginary part of the refractive index is especially 772
large [264,265]. Soon afterwards, a QED description for the rate of RET in the presence of 773
dispersing and absorbing material bodies of arbitrary shapes was provided [266]. In the 21st century, 774
among other advances, quantum theory has helped us understand the role of mediators in energy 775
transfer (i.e. 3- and 4-body RET) and the rederivations of the RET coupling tensor has provided new 776
physical insights. 777
In the last ten years, research into RET has moved into many exciting directions – too numerous to 778
cover in detail in a single review. For example, the enhancement and control of long-range, super-779
Coulombic RET in hyperbolic metamaterials is shown [267,268] and the influence of epsilon-and-780
mu-near-zero waveguide super-coupling on RET is considered [269]. Moreover, many research 781
groups continue to unravel the nature of energy transfer within biological photosynthesis, with a 782
special focus on the understanding of the roles that molecular vibrations may play in facilitating the 783
process. There are also enormous efforts to develop ‘energy materials’ that may enable new 784
technologies, which include those focused on solar energy harvesting. Materials based on surface 785
plasmons have shown great promise, especially in its connection to the huge enhancements of RET 786
efficiency. Research groups are also working on RET in both the non-optical regions of the 787
electromagnetic spectrum and within optical cavities. In all of these exciting areas of research, new 788
experiments and theory need continued development. The theory of QED, while the most precise 789
theory we know for light-matter interactions, assumes non-dissipative closed systems and that the 790
electrons are localised to the molecules. Consequently, in its current formulation, microscopic QED 791
is not directly applicable to the investigation of surface plasmons (delocalised excitons) or the 792
process of decoherence, which occurs because the system is open to the environment. While semi-793
classical theories can address these questions in a limited way, the continued development of 794
macroscopic QED [270] is desirable for accurate portrayals of such processes. 795
Jones and Bradshaw RET: Theory to Applications
28
This is a provisional file, not the final typeset article
Conflict of Interest 796
The authors declare that the research was conducted in the absence of any commercial or financial 797
relationships that could be construed as a potential conflict of interest. 798
799
Author Contributions 800
GJ and DB equally wrote and edited the final manuscript. The figures and table are produced by 801
them. 802
803
Funding 804
This research received no external funding. 805
806
Acknowledgments 807
We are grateful for many helpful comments from Professor David Andrews, Dr. Kayn Forbes and 808 Dr. Stefan Buhmann. 809
810
References 811
1. van der Meer BW, Coker G, Chen SYS. Resonance Energy Transfer: Theory and Data. 812
New York: VCH (1994). 813
2. Andrews DL, Demidov AA. Resonance Energy Transfer. Chichester: Wiley (1999). 814
3. May V. Charge and energy transfer dynamics in molecular systems. Hoboken, NJ: John 815
Wiley & Sons (2008). 816
4. Medintz I, Hildebrandt N. Förster Resonance Energy Transfer: From Theory to 817
Applications. Weinheim: Wiley-VCH (2013). 818
5. Dexter DL. A theory of sensitized luminescence in solids. J. Chem. Phys. (1953) 21:836-819
850. doi: 10.1063/1.1699044 820
6. Franck J. Einige aus der Theorie von Klein und Bosseland zu ziehende Folgerungen über 821
Fluoreszenz, photochemische Prozesse und die Elektronenemission glühender Körper. Z. 822
Phys. (1922) 9:259-266. doi: 10.1007/bf01326976 823
Jones and Bradshaw RET: Theory to Applications
29
7. Carlo G. Über Entstehung wahrer Lichtabsorption und scheinbare Koppelung von 824
Quantensprüngen. Z. Phys. (1922) 10:185-199. doi: 10.1007/bf01332559 825
8. Cario G, Franck J. Über Zerlegung von Wasserstoffmolekülen durch angeregte 826
Quecksilberatome. Z. Phys. (1922) 11:161-166. doi: 10.1007/bf01328410 827
9. Perrin J. Fluorescence et induction moléculaire par résonance. C. R. Acad. Sci. (1927) 828
184:1097-1100. 829
10. Perrin F. Théorie quantique des transferts d’activation entre molécules de même espèce. Cas 830
des solutions fluorescentes. Ann. Phys. (Berlin) (1932) 10:283-314. doi: 831
10.1051/anphys/193210170283 832
11. Kallmann H, London F. Über quantenmechanische Energieübertragung zwischen atomaren 833
Systemen. Z. Phys. Chem. (1929) 2B:207-243 doi: 10.1515/zpch-1929-0214 834
12. Förster T. Energiewanderung und Fluoreszenz. Naturwissenschaften (1946) 33:166-175. 835
doi: 10.1007/bf00585226 836
13. Förster T. Zwischenmolekulare Energiewanderung und Fluoreszenz. Ann. Phys. (Berlin) 837
(1948) 437:55-75. doi: 10.1002/andp.19484370105 838
14. Förster T. 10th Spiers Memorial Lecture. Transfer mechanisms of electronic excitation. 839
Discuss. Faraday Soc. (1959) 27:7-17. doi: 10.1039/DF9592700007 840
15. Latt SA, Cheung HT, Blout ER. Energy Transfer. A system with relatively fixed donor-841
acceptor separation. J. Am. Chem. Soc. (1965) 87:995-1003. doi: 10.1021/ja01083a011 842
16. Stryer L, Haugland RP. Energy transfer: a spectroscopic ruler. Proc. Natl. Acad. Sci. USA 843
(1967) 58:719-726. doi: 10.1073/pnas.58.2.719 844
17. Sahoo H. Förster resonance energy transfer – A spectroscopic nanoruler: Principle and 845
applications. J. Photochem. Photobiol. C (2011) 12:20-30. doi: 846
10.1016/j.jphotochemrev.2011.05.001 847
18. Ruggenthaler M, Tancogne-Dejean N, Flick J, Appel H, Rubio A. From a quantum-848
electrodynamical light–matter description to novel spectroscopies. Nat. Rev. Chem. (2018) 849
2:0118. doi: 10.1038/s41570-018-0118 850
19. Dirac PAM. The quantum theory of the emission and absorption of radiation. Proc. R. Soc. 851
A (1927) 114:243-265. doi: 10.1098/rspa.1927.0039 852
20. Dirac PAM. The Principles of Quantum Mechanics, Fourth Edition. Oxford: Clarendon 853
Press (1981). 854
21. Feynman RP. Relativistic cut-off for quantum electrodynamics. Phys. Rev. (1948) 74:1430-855
1438. doi: 10.1103/PhysRev.74.1430 856
Jones and Bradshaw RET: Theory to Applications
30
This is a provisional file, not the final typeset article
22. Feynman RP. The theory of positrons. Phys. Rev. (1949) 76:749-759. doi: 857
10.1103/PhysRev.76.749 858
23. Feynman RP. Space-time approach to quantum electrodynamics. Phys. Rev. (1949) 76:769-859
789. doi: 10.1103/PhysRev.76.769 860
24. Feynman RP. Mathematical formulation of the quantum theory of electromagnetic 861
interaction. Phys. Rev. (1950) 80:440-457. doi: 10.1103/PhysRev.80.440 862
25. Feynman RP. An operator calculus having applications in quantum electrodynamics. Phys. 863
Rev. (1951) 84:108-128. doi: 10.1103/PhysRev.84.108 864
26. Schwinger J. On quantum-electrodynamics and the magnetic moment of the electron. Phys. 865
Rev. (1948) 73:416-417. doi: 10.1103/PhysRev.73.416 866
27. Schwinger J. Quantum electrodynamics. I. A covariant formulation. Phys. Rev. (1948) 867
74:1439-1461. doi: 10.1103/PhysRev.74.1439 868
28. Schwinger J. Quantum electrodynamics. II. Vacuum polarization and self-energy. Phys. Rev. 869
(1949) 75:651-679. doi: 10.1103/PhysRev.75.651 870
29. Schwinger J. Quantum electrodynamics. III. The electromagnetic properties of the electron-871
radiative corrections to scattering. Phys. Rev. (1949) 76:790-817. doi: 872
10.1103/PhysRev.76.790 873
30. Tomonaga S-I, Oppenheimer JR. On infinite field reactions in quantum field theory. Phys. 874
Rev. (1948) 74:224-225. doi: 10.1103/PhysRev.74.224 875
31. Tomonaga S-I. Development of quantum electrodynamics. Science (1966) 154:864-868. doi: 876
10.1126/science.154.3751.864 877
32. Dyson FJ. The radiation theories of Tomonaga, Schwinger, and Feynman. Phys. Rev. (1949) 878
75:486-502. doi: 10.1103/PhysRev.75.486 879
33. Hanneke D, Fogwell Hoogerheide S, Gabrielse G. Cavity control of a single-electron 880
quantum cyclotron: Measuring the electron magnetic moment. Phys. Rev. A (2011) 881
83:052122. doi: 10.1103/PhysRevA.83.052122 882
34. Aoyama T, Kinoshita T, Nio M. Revised and improved value of the QED tenth-order 883
electron anomalous magnetic moment. Phys. Rev. D (2018) 97:036001. doi: 884
10.1103/PhysRevD.97.036001 885
35. Pachucki K. Complete two-loop binding correction to the Lamb shift. Phys. Rev. Lett. 886
(1994) 72:3154-3157. doi: 10.1103/PhysRevLett.72.3154 887
Jones and Bradshaw RET: Theory to Applications
31
36. Hagley EW, Pipkin FM. Separated oscillatory field measurement of hydrogen 2S1/2-2P3/2 888
fine structure interval. Phys. Rev. Lett. (1994) 72:1172-1175. doi: 889
10.1103/PhysRevLett.72.1172 890
37. Casimir HBG, Polder D. The influence of retardation on the London-van der Waals forces. 891
Phys. Rev. (1948) 73:360-372. doi: 10.1103/PhysRev.73.360 892
38. Buhmann SY, Knöll L, Welsch D-G, Dung HT. Casimir-Polder forces: A nonperturbative 893
approach. Phys. Rev. A (2004) 70:052117. doi: 10.1103/PhysRevA.70.052117 894
39. Przybytek M, Jeziorski B, Cencek W, Komasa J, Mehl JB, Szalewicz K. Onset of Casimir-895
Polder retardation in a long-range molecular quantum state. Phys. Rev. Lett. (2012) 896
108:183201. doi: 10.1103/PhysRevLett.108.183201 897
40. Salam A. Non-Relativistic QED Theory of the van der Waals Dispersion Interaction. Cham, 898
Switzerland: Springer (2016). 899
41. Passante R. Dispersion interactions between neutral atoms and the quantum 900
electrodynamical vacuum. (2018) 10:735. doi: 10.3390/atoms6040056 901
42. Jackson JD. Classical electrodynamics. New York: Wiley (1975). 902
43. Power EA, Zienau S. Coulomb gauge in non-relativistic quantum electrodynamics and the 903
shape of spectral lines. Philos. Trans. R. Soc. A (1959) 251:427-454. doi: 904
10.1098/rsta.1959.0008 905
44. Woolley R. Molecular quantum electrodynamics. Proc. R. Soc. A (1971) 321:557-572. doi: 906
10.1098/rspa.1971.0049 907
45. Craig DP, Thirunamachandran T. Molecular Quantum Electrodynamics: An Introduction to 908
Radiation-Molecule Interactions. Mineola, NY: Dover Publications (1998). 909
46. Salam A. Molecular quantum electrodynamics in the Heisenberg picture: A field theoretic 910
viewpoint. Int. Rev. Phys. Chem. (2008) 27:405-448. doi: 10.1080/01442350802045206 911
47. Salam A. Molecular Quantum Electrodynamics. Long-Range Intermolecular Interactions. 912
Hoboken, NJ: Wiley (2010). 913
48. Andrews DL, Jones GA, Salam A, Woolley RG. Quantum Hamiltonians for optical 914
interactions. J. Chem. Phys. (2018) 148:040901. doi: 10.1063/1.5018399 915
49. Andrews DL, Bradshaw DS. Virtual photons, dipole fields and energy transfer: a quantum 916
electrodynamical approach. Eur. J. Phys. (2004) 25:845-858. doi: 10.1088/0143-917
0807/25/6/017 918
Jones and Bradshaw RET: Theory to Applications
32
This is a provisional file, not the final typeset article
50. Lock MPE, Andrews DL, Jones GA. On the nature of long range electronic coupling in a 919
medium: Distance and orientational dependence for chromophores in molecular aggregates. 920
J. Chem. Phys. (2014) 140:044103. doi: 10.1063/1.4861695 921
51. Frost JE, Jones GA. A quantum dynamical comparison of the electronic couplings derived 922
from quantum electrodynamics and Förster theory: application to 2D molecular aggregates. 923
New J. Phys. (2014) 16:113067. doi: 10.1088/1367-2630/16/11/113067 924
52. Avery JS. The retarded dipole-dipole interaction in exciton theory. Proc. Phys. Soc. (1966) 925
89:677-682. doi: 10.1088/0370-1328/89/3/321 926
53. Wheeler JA, Feynman RP. Classical electrodynamics in terms of direct interparticle action. 927
Rev. Mod. Phys. (1949) 21:425-433. doi: 10.1103/RevModPhys.21.425 928
54. Gomberoff L, Power EA. The resonance transfer of excitation. Proc. Phys. Soc. (1966) 929
88:281-284. doi: 10.1088/0370-1328/88/2/302 930
55. Power EA, Thirunamachandran T. Quantum electrodynamics with nonrelativistic sources. I. 931
Transformation to the multipolar formalism for second-quantized electron and Maxwell 932
interacting fields. Phys. Rev. A (1983) 28:2649-2662. doi: 10.1103/PhysRevA.28.2649 933
56. Power EA, Thirunamachandran T. Quantum electrodynamics with nonrelativistic sources. 934
II. Maxwell fields in the vicinity of a molecule. Phys. Rev. A (1983) 28:2663-2670. doi: 935
10.1103/PhysRevA.28.2663 936
57. Power EA, Thirunamachandran T. Quantum electrodynamics with nonrelativistic sources. 937
III. Intermolecular interactions. Phys. Rev. A (1983) 28:2671-2675. doi: 938
10.1103/PhysRevA.28.2671 939
58. Einstein A. Die Grundlage der allgemeinen Relativitätstheorie. Ann. Phys. (Berlin) (1916) 940
354:769-822. doi: 10.1002/andp.19163540702 941
59. Craig DP, Thirunamachandran T. Radiation-molecule interactions in chemical physics. Adv. 942
Quant. Chem. (1982) 16:97-160. doi: 10.1016/S0065-3276(08)60352-4 943
60. Newton RG. Scattering Theory of Waves and Particles. New York, Heidelberg, Berlin: 944
Springer-Verlag (2013). 945
61. Andrews DL, Sherborne BS. Resonant excitation transfer: A quantum electrodynamical 946
study. J. Chem. Phys. (1987) 86:4011-4017. doi: 10.1063/1.451910 947
62. Andrews DL. A unified theory of radiative and radiationless molecular energy transfer. 948
Chem. Phys. (1989) 135:195-201. doi: 10.1016/0301-0104(89)87019-3 949
63. Daniels GJ, Jenkins RD, Bradshaw DS, Andrews DL. Resonance energy transfer: The 950
unified theory revisited. J. Chem. Phys. (2003) 119:2264-2274. doi: 10.1063/1.1579677 951
Jones and Bradshaw RET: Theory to Applications
33
64. Andrews DL, Juzeliūnas. G. Intermolecular energy transfer: Retardation effects. J. Chem. 952
Phys. (1992) 96:6606-6612. doi: 10.1063/1.462599 953
65. Jenkins RD, Daniels GJ, Andrews DL. Quantum pathways for resonance energy transfer. J. 954
Chem. Phys. (2004) 120:11442-11448. doi: 10.1063/1.1742697 955
66. Grinter R, Jones GA. Resonance energy transfer: The unified theory via vector spherical 956
harmonics. J. Chem. Phys. (2016) 145:074107. doi: 10.1063/1.4960732 957
67. Jones GA, Grinter R. The plane- and spherical-wave descriptions of electromagnetic 958
radiation: a comparison and discussion of their relative merits. Eur. J. Phys. (2018) 959
39:053001. doi: 10.1088/1361-6404/aac366 960
68. Grinter R, Jones GA. Interpreting angular momentum transfer between electromagnetic 961
multipoles using vector spherical harmonics. Opt. Lett. (2018) 43:367-370. doi: 962
10.1364/OL.43.000367 963
69. Scholes GD, Andrews DL. Damping and higher multipole effects in the quantum 964
electrodynamical model for electronic energy transfer in the condensed phase. J. Chem. 965
Phys. (1997) 107:5374-5384. doi: 10.1063/1.475145 966
70. Salam A. Resonant transfer of excitation between two molecules using Maxwell fields. J. 967
Chem. Phys. (2005) 122:044113. doi: 10.1063/1.1827596 968
71. Salam A. A general formula for the rate of resonant transfer of energy between two electric 969
multipole moments of arbitrary order using molecular quantum electrodynamics. J. Chem. 970
Phys. (2005) 122:044112. doi: 10.1063/1.1830430 971
72. Andrews DL. On the conveyance of angular momentum in electronic energy transfer. Phys. 972
Chem. Chem. Phys. (2010) 12:7409-7417. doi: 10.1039/c002313m 973
73. Andrews DL. Optical angular momentum: Multipole transitions and photonics. Phys. Rev. A 974
(2010) 81:033825. doi: 10.1103/PhysRevA.81.033825 975
74. Rice EM, Bradshaw DS, Saadi K, Andrews DL. Identifying the development in phase and 976
amplitude of dipole and multipole radiation. Eur. J. Phys. (2012) 33:345-358. doi: 977
10.1088/0143-0807/33/2/345 978
75. Loudon R. The Quantum Theory of Light. Oxford: Oxford University Press (2000). 979
76. Andrews DL, Forbes KA. Quantum features in the orthogonality of optical modes for 980
structured and plane-wave light. Opt. Lett. (2018) 43:3249-3252. doi: 981
10.1364/OL.43.003249 982
77. Salam A. The unified theory of resonance energy transfer according to molecular quantum 983
electrodynamics. Atoms (2018) 6:56. doi: 10.3390/atoms6040056 984
Jones and Bradshaw RET: Theory to Applications
34
This is a provisional file, not the final typeset article
78. Fermi E. Nuclear Physics. Chicago: University of Chicago Press (1950). 985
79. Andrews DL, Thirunamachandran T. On three-dimensional rotational averages. J. Chem. 986
Phys. (1977) 67:5026-5033. doi: 10.1063/1.434725 987
80. Förster T. "Mechanisms of Energy Transfer". In: Florkin M and Stotz EH, editors. 988
Comprehensive Biochemistry, Amsterdam: Elsevier (1967). p. 61-80. 989
81. Andrews DL, Bradshaw DS. The role of virtual photons in nanoscale photonics. Ann. Phys. 990
(Berlin) (2014) 526:173-186. doi: 10.1002/andp.201300219 991
82. Nasiri Avanaki K, Ding W, Schatz GC. Resonance energy transfer in arbitrary media: 992
Beyond the point dipole approximation. J. Phys. Chem. C (2018) 122:29445-29456. doi: 993
10.1021/acs.jpcc.8b07407 994
83. Craig DP, Power EA, Thirunamachandran T. The interaction of optically active molecules. 995
Proc. R. Soc. A (1971) 322:165-179. doi: 10.1098/rspa.1971.0061 996
84. Craig D, Thirunamachandran T. Chiral discrimination in molecular excitation transfer. J. 997
Chem. Phys. (1998) 109:1259-1263. doi: 10.1063/1.476676 998
85. Rodriguez JJ, Salam A. Effect of medium chirality on the rate of resonance energy transfer. 999
J. Phys. Chem. B (2010) 115:5183-5190. doi: 10.1021/jp105715z 1000
86. Andrews DL. Chirality in fluorescence and energy transfer. Methods Appl. Fluoresc. (2019) 1001
7:032001. doi: 10.1088/2050-6120/ab10f0 1002
87. Salam A. Mediation of resonance energy transfer by a third molecule. J. Chem. Phys. (2012) 1003
136:014509. doi: 10.1063/1.3673779 1004
88. Andrews DL, Ford JS. Resonance energy transfer: Influence of neighboring matter 1005
absorbing in the wavelength region of the acceptor. J. Chem. Phys. (2013) 139:014107. doi: 1006
10.1063/1.4811793 1007
89. Ford JS, Andrews DL. Geometrical effects on resonance energy transfer between 1008
orthogonally-oriented chromophores, mediated by a nearby polarisable molecule. Chem. 1009
Phys. Lett. (2014) 591:88-92. doi: 10.1016/j.cplett.2013.11.002 1010
90. Weeraddana D, Premaratne M, Andrews DL. Direct and third-body mediated resonance 1011
energy transfer in dimensionally constrained nanostructures. Phys. Rev. B (2015) 92:035128. 1012
doi: 10.1103/PhysRevB.92.035128 1013
91. Salam A. Near-zone mediation of RET by one and two proximal particles. J. Phys. Chem. A 1014
(2019) 123:2853-2860. doi: 10.1021/acs.jpca.9b00827 1015
Jones and Bradshaw RET: Theory to Applications
35
92. Qian H, Georgi C, Anderson N, Green AA, Hersam MC, Novotny L, Hartschuh A. Exciton 1016
energy transfer in pairs of single-walled carbon nanotubes. Nano Lett. (2008) 8:1363-1367. 1017
doi: 10.1021/nl080048r 1018
93. Lefebvre J, Finnie P. Photoluminescence and Förster resonance energy transfer in elemental 1019
bundles of single-walled carbon nanotubes. J. Phys. Chem. C (2009) 113:7536-7540. doi: 1020
10.1021/jp810892z 1021
94. Wong CY, Curutchet C, Tretiak S, Scholes GD. Ideal dipole approximation fails to predict 1022
electronic coupling and energy transfer between semiconducting single-wall carbon 1023
nanotubes. J. Chem. Phys. (2009) 130:081104. doi: 10.1063/1.3088846 1024
95. Mehlenbacher RD, McDonough TJ, Grechko M, Wu M-Y, Arnold MS, Zanni MT. Energy 1025
transfer pathways in semiconducting carbon nanotubes revealed using two-dimensional 1026
white-light spectroscopy. Nat. Commun. (2015) 6:6732. doi: 10.1038/ncomms7732 1027
96. Davoody AH, Karimi F, Arnold MS, Knezevic I. Theory of exciton energy transfer in 1028
carbon nanotube composites. J. Phys. Chem. C (2016) 120:16354-16366. doi: 1029
10.1021/acs.jpcc.6b04050 1030
97. Kagan CR, Murray CB, Nirmal M, Bawendi MG. Electronic energy transfer in CdSe 1031
quantum dot solids. Phys. Rev. Lett. (1996) 76:1517-1520. doi: 1032
10.1103/PhysRevLett.76.1517 1033
98. Koole R, Liljeroth P, de Mello Donegá C, Vanmaekelbergh D, Meijerink A. Electronic 1034
coupling and exciton energy transfer in CdTe quantum-dot molecules. J. Am. Chem. Soc. 1035
(2006) 128:10436-10441. doi: 10.1021/ja061608w 1036
99. Clark SW, Harbold JM, Wise FW. Resonant energy transfer in PbS quantum dots. J. Phys. 1037
Chem. C (2007) 111:7302-7305. doi: 10.1021/jp0713561 1038
100. Bruchez M, Moronne M, Gin P, Weiss S, Alivisatos AP. Semiconductor nanocrystals as 1039
fluorescent biological labels. Science (1998) 281:2013-2016. doi: 1040
10.1126/science.281.5385.2013 1041
101. Chan WCW, Nie S. Quantum dot bioconjugates for ultrasensitive nonisotopic detection. 1042
Science (1998) 281:2016-2018. doi: 10.1126/science.281.5385.2016 1043
102. Clapp AR, Medintz IL, Mattoussi H. Förster resonance energy transfer investigations using 1044
quantum-dot fluorophores. ChemPhysChem (2006) 7:47-57. doi: 10.1002/cphc.200500217 1045
103. Medintz IL, Mattoussi H. Quantum dot-based resonance energy transfer and its growing 1046
application in biology. Phys. Chem. Chem. Phys. (2009) 11:17-45. doi: 10.1039/B813919A 1047
Jones and Bradshaw RET: Theory to Applications
36
This is a provisional file, not the final typeset article
104. Willard DM, Van Orden A. Resonant energy-transfer sensor. Nat. Mater. (2003) 2:575-576. 1048
doi: 10.1038/nmat972 1049
105. Sapsford KE, Granek J, Deschamps JR, Boeneman K, Blanco-Canosa JB, Dawson PE, 1050
Susumu K, Stewart MH, Medintz IL. Monitoring botulinum neurotoxin A activity with 1051
peptide-functionalized quantum dot resonance energy transfer sensors. ACS Nano (2011) 1052
5:2687-2699. doi: 10.1021/nn102997b 1053
106. Algar WR, Ancona MG, Malanoski AP, Susumu K, Medintz IL. Assembly of a concentric 1054
Förster resonance energy transfer relay on a quantum dot scaffold: Characterization and 1055
application to multiplexed protease sensing. ACS Nano (2012) 6:11044-11058. doi: 1056
10.1021/nn304736j 1057
107. Chou KF, Dennis AM. Förster resonance energy transfer between quantum dot donors and 1058
quantum dot acceptors. Sensors (2015) 15:13288-13325. doi: 10.3390/s150613288 1059
108. Qiu X, Hildebrandt N. Rapid and multiplexed microRNA diagnostic assay using quantum 1060
dot-based Förster resonance energy transfer. ACS Nano (2015) 9:8449-8457. doi: 1061
10.1021/acsnano.5b03364 1062
109. Stanisavljevic M, Krizkova S, Vaculovicova M, Kizek R, Adam V. Quantum dots-1063
fluorescence resonance energy transfer-based nanosensors and their application. Biosens. 1064
Bioelectron. (2015) 74:562-574. doi: 10.1016/j.bios.2015.06.076 1065
110. Shi J, Tian F, Lyu J, Yang M. Nanoparticle based fluorescence resonance energy transfer 1066
(FRET) for biosensing applications. J. Mater. Chem. B (2015) 3:6989-7005. doi: 1067
10.1039/C5TB00885A 1068
111. Hildebrandt N, Spillmann CM, Algar WR, Pons T, Stewart MH, Oh E, Susumu K, Díaz SA, 1069
Delehanty JB, Medintz IL. Energy transfer with semiconductor quantum dot bioconjugates: 1070
A versatile platform for biosensing, energy harvesting, and other developing applications. 1071
Chem. Rev. (2017) 117:536-711. doi: 10.1021/acs.chemrev.6b00030 1072
112. Samia ACS, Dayal S, Burda C. Quantum dot-based energy transfer: Perspectives and 1073
potential for applications in photodynamic therapy. Photochem. Photobiol. (2006) 82:617-1074
625. doi: 10.1562/2005-05-11-ir-525 1075
113. Li L, Zhao J-F, Won N, Jin H, Kim S, Chen J-Y. Quantum dot-aluminum phthalocyanine 1076
conjugates perform photodynamic reactions to kill cancer cells via fluorescence resonance 1077
energy transfer. Nanoscale Res. Lett. (2012) 7:386. doi: 10.1186/1556-276x-7-386 1078
114. Scholes GD, Andrews DL. Resonance energy transfer and quantum dots. Phys. Rev. B 1079
(2005) 72:125331. doi: 10.1103/PhysRevB.72.125331 1080
Jones and Bradshaw RET: Theory to Applications
37
115. Allan G, Delerue C. Energy transfer between semiconductor nanocrystals: Validity of 1081
Förster's theory. Phys. Rev. B (2007) 75:195311. doi: 10.1103/PhysRevB.75.195311 1082
116. Curutchet C, Franceschetti A, Zunger A, Scholes GD. Examining Förster energy transfer for 1083
semiconductor nanocrystalline quantum dot donors and acceptors. J. Phys. Chem. C (2008) 1084
112:13336-13341. doi: 10.1021/jp805682m 1085
117. Weeraddana D, Premaratne M, Gunapala SD, Andrews DL. Quantum electrodynamical 1086
theory of high-efficiency excitation energy transfer in laser-driven nanostructure systems. 1087
Phys. Rev. B (2016) 94:085133. doi: 10.1103/PhysRevB.94.085133 1088
118. Weeraddana D, Premaratne M, Andrews DL. Quantum electrodynamics of resonance energy 1089
transfer in nanowire systems. Phys. Rev. B (2016) 93:075151. doi: 1090
10.1103/PhysRevB.93.075151 1091
119. Moroz P, Royo Romero L, Zamkov M. Colloidal semiconductor nanocrystals in energy 1092
transfer reactions. Chem. Commun. (2019) 55:3033-3048. doi: 10.1039/C9CC00162J 1093
120. Liu X, Qiu J. Recent advances in energy transfer in bulk and nanoscale luminescent 1094
materials: from spectroscopy to applications. Chem. Soc. Rev. (2015) 44:8714-8746. doi: 1095
10.1039/C5CS00067J 1096
121. Nabiev I, Rakovich A, Sukhanova A, Lukashev E, Zagidullin V, Pachenko V, Rakovich YP, 1097
Donegan JF, Rubin AB, Govorov AO. Fluorescent quantum dots as artificial antennas for 1098
enhanced light harvesting and energy transfer to photosynthetic reaction centers. Angew. 1099
Chem. Int. Ed. (2010) 49:7217-7221. doi: 10.1002/anie.201003067 1100
122. Adronov A, Fréchet JMJ. Light-harvesting dendrimers. Chem. Commun. (2000):1701-1710. 1101
doi: 10.1039/B005993P 1102
123. Balzani V, Ceroni P, Maestri M, Vicinelli V. Light-harvesting dendrimers. Curr. Opin. 1103
Chem. Biol. (2003) 7:657-665. doi: 10.1016/j.cbpa.2003.10.001 1104
124. Nantalaksakul A, Reddy DR, Bardeen CJ, Thayumanavan S. Light harvesting dendrimers. 1105
Photosynth. Res. (2006) 87:133-150. doi: 10.1007/s11120-005-8387-3 1106
125. Ziessel R, Ulrich G, Haefele A, Harriman A. An artificial light-harvesting array constructed 1107
from multiple bodipy dyes. J. Am. Chem. Soc. (2013) 135:11330-11344. doi: 1108
10.1021/ja4049306 1109
126. Zhang X, Zeng Y, Yu T, Chen J, Yang G, Li Y. Advances in photofunctional dendrimers for 1110
solar energy conversion. J. Phys. Chem. Lett. (2014) 5:2340-2350. doi: 10.1021/jz5007862 1111
127. Pan S-J, Ma D-D, Liu J-S, Chen K-Z, Zhang T-T, Peng Y-R. Benzophenone and zinc(II) 1112
phthalocyanine dichromophores-labeled poly (aryl ether) dendrimer: synthesis, 1113
Jones and Bradshaw RET: Theory to Applications
38
This is a provisional file, not the final typeset article
characterization and photoinduced energy transfer. J. Coord. Chem. (2016) 69:618-627. doi: 1114
10.1080/00958972.2015.1131271 1115
128. Zou Q, Liu K, Abbas M, Yan X. Peptide-modulated self-assembly of chromophores toward 1116
biomimetic light-harvesting nanoarchitectonics. Adv. Mater. (2016) 28:1031-1043. doi: 1117
10.1002/adma.201502454 1118
129. Nelson T, Fernandez-Alberti S, Roitberg AE, Tretiak S. Electronic delocalization, 1119
vibrational dynamics, and energy transfer in organic chromophores. J. Phys. Chem. Lett. 1120
(2017) 8:3020-3031. doi: 10.1021/acs.jpclett.7b00790 1121
130. Viswanath V, Santhakumar K. Perspectives on dendritic architectures and their biological 1122
applications: From core to cell. Journal of Photochemistry and Photobiology (2017) 173:61-1123
83. doi: 10.1016/j.jphotobiol.2017.05.023 1124
131. Jenkins RD, Andrews DL. Multichromophore excitons and resonance energy transfer: 1125
Molecular quantum electrodynamics. J. Chem. Phys. (2003) 118:3470-3479. doi: 1126
10.1063/1.1538611 1127
132. Andrews DL, Bradshaw DS. Optically nonlinear energy transfer in light-harvesting 1128
dendrimers. J. Chem. Phys. (2004) 121:2445-2454. doi: 10.1063/1.1769354 1129
133. Andrews DL, Li SP, Rodríguez J, Slota J. Development of the energy flow in light-1130
harvesting dendrimers. J. Chem. Phys. (2007) 127. doi: 10.1063/1.2785175 1131
134. May V. Beyond the Förster theory of excitation energy transfer: importance of higher-order 1132
processes in supramolecular antenna systems. Dalton Trans. (2009):10086-10105. doi: 1133
10.1039/B908567J 1134
135. Olaya-Castro A, Scholes GD. Energy transfer from Förster-Dexter theory to quantum 1135
coherent light-harvesting. Int. Rev. Phys. Chem. (2011) 30:49-77. doi: 1136
10.1080/0144235x.2010.537060 1137
136. Jang S. Theory of multichromophoric coherent resonance energy transfer: A polaronic 1138
quantum master equation approach. J. Chem. Phys. (2011) 135:034105. doi: 1139
10.1063/1.3608914 1140
137. Bradshaw DS, Andrews DL. Mechanisms of light energy harvesting in dendrimers and 1141
hyperbranched polymers. Polymers (2011) 3:2053-2077. doi: 10.3390/polym3042053 1142
138. Schröter M, Ivanov SD, Schulze J, Polyutov SP, Yan Y, Pullerits T, Kühn O. Exciton-1143
vibrational coupling in the dynamics and spectroscopy of Frenkel excitons in molecular 1144
aggregates. Phys. Rep. (2015) 567:1-78. doi: 10.1016/j.physrep.2014.12.001 1145
Jones and Bradshaw RET: Theory to Applications
39
139. Freixas VM, Ondarse-Alvarez D, Tretiak S, Makhov DV, Shalashilin DV, Fernandez-1146
Alberti S. Photoinduced non-adiabatic energy transfer pathways in dendrimer building 1147
blocks. J. Chem. Phys. (2019) 150:124301. doi: 10.1063/1.5086680 1148
140. Workineh ZG. Effect of surface functionalization on the structural properties of single 1149
dendrimers: Monte Carlo simulation study. Comput. Mater. Sci. (2019) 168:40-47. doi: 1150
10.1016/j.commatsci.2019.05.061 1151
141. Jenkins RD, Andrews DL. Three-center systems for energy pooling: Quantum 1152
electrodynamical theory. J. Phys. Chem. A (1998) 102:10834-10842. doi: 1153
10.1021/jp983071h 1154
142. Andrews DL, Curutchet C, Scholes GD. Resonance energy transfer: Beyond the limits. 1155
Laser & Photon. Rev. (2011) 5:114-123. doi: 10.1002/lpor.201000004 1156
143. Steer RP. Electronic energy pooling in organic systems: a cross-disciplinary tutorial review. 1157
Can. J. Chem. (2017) 95:1025-1040. doi: 10.1139/cjc-2017-0369 1158
144. Zhang C, Sun L, Zhang Y, Yan C. Rare earth upconversion nanophosphors: synthesis, 1159
functionalization and application as biolabels and energy transfer donors. J. Rare Earth. 1160
(2010) 28:807-819. doi: 10.1016/S1002-0721(09)60206-4 1161
145. Dong H, Sun L-D, Yan C-H. Energy transfer in lanthanide upconversion studies for 1162
extended optical applications. Chem. Soc. Rev. (2015) 44:1608-1634. doi: 1163
10.1039/C4CS00188E 1164
146. Weingarten DH, LaCount MD, van de Lagemaat J, Rumbles G, Lusk MT, Shaheen SE. 1165
Experimental demonstration of photon upconversion via cooperative energy pooling. Nat. 1166
Commun. (2017) 8:14808. doi: 10.1038/ncomms14808 1167
147. LaCount MD, Weingarten D, Hu N, Shaheen SE, van de Lagemaat J, Rumbles G, Walba 1168
DM, Lusk MT. Energy pooling upconversion in organic molecular systems. J. Phys. Chem. 1169
A (2015) 119:4009-4016. doi: 10.1021/acs.jpca.5b00509 1170
148. LaCount MD, Lusk MT. Electric dipole coupling in optical cavities and its implications for 1171
energy transfer, up-conversion, and pooling. Phys. Rev. A (2016) 93:063811. doi: 1172
10.1103/PhysRevA.93.063811 1173
149. LaCount MD, Lusk MT. Improved energy pooling efficiency through inhibited spontaneous 1174
emission. J. Phys. Chem. C (2017) 121:8335-8344. doi: 10.1021/acs.jpcc.7b01693 1175
150. LaCount MD, Lusk MT. Quantum cutting using organic molecules. Phys. Chem. Chem. 1176
Phys. (2019) 21:7814-7821. doi: 10.1039/C9CP00329K 1177
Jones and Bradshaw RET: Theory to Applications
40
This is a provisional file, not the final typeset article
151. Allcock P, Andrews DL. Two-photon fluorescence: Resonance energy transfer. J. Chem. 1178
Phys. (1998) 108:3089-3095. doi: 10.1063/1.475706 1179
152. Bradshaw DS, Andrews DL. Competing mechanisms for energy transfer in two-photon 1180
absorbing systems. Chem. Phys. Lett. (2006) 430:191-194. doi: 10.1016/j.cplett.2006.08.116 1181
153. Dichtel WR, Serin JM, Edder C, Fréchet JMJ, Matuszewski M, Tan L-S, Ohulchanskyy TY, 1182
Prasad PN. Singlet oxygen generation via two-photon excited FRET. J. Am. Chem. Soc. 1183
(2004) 126:5380-5381. doi: 10.1021/ja031647x 1184
154. Oar MA, Serin JM, Dichtel WR, Fréchet JMJ, Ohulchanskyy TY, Prasad PN. 1185
Photosensitization of singlet oxygen via two-photon-excited fluorescence resonance energy 1186
transfer in a water-soluble dendrimer. Chem. Mater. (2005) 17:2267-2275. doi: 1187
10.1021/cm047825i 1188
155. Tian N, Xu Q-H. Enhanced two-photon excitation fluorescence by fluorescence resonance 1189
energy transfer using conjugated polymers. Adv. Mater. (2007) 19:1988-1991. doi: 1190
10.1002/adma.200700654 1191
156. Cheng S-H, Hsieh C-C, Chen N-T, Chu C-H, Huang C-M, Chou P-T, Tseng F-G, Yang C-S, 1192
Mou C-Y, Lo L-W. Well-defined mesoporous nanostructure modulates three-dimensional 1193
interface energy transfer for two-photon activated photodynamic therapy. Nano Today 1194
(2011) 6:552-563. doi: 10.1016/j.nantod.2011.10.003 1195
157. Ngen EJ, Xiao L, Rajaputra P, Yan X, You Y. Enhanced singlet oxygen generation from a 1196
porphyrin–rhodamine B dyad by two-photon excitation through resonance energy transfer. 1197
Photochem. Photobiol. (2013) 89:841-848. doi: 10.1111/php.12071 1198
158. Chen N-T, Tang K-C, Chung M-F, Cheng S-H, Huang C-M, Chu C-H, Chou P-T, Souris JS, 1199
Chen C-T, Mou C-Y, Lo L-W. Enhanced plasmonic resonance energy transfer in 1200
mesoporous silica-encased gold nanorod for two-photon-activated photodynamic therapy. 1201
Theranostics (2014) 4:798-807. doi: 10.7150/thno.8934 1202
159. Drozdek S, Szeremeta J, Lamch L, Nyk M, Samoc M, Wilk KA. Two-photon induced 1203
fluorescence energy transfer in polymeric nanocapsules containing CdSexS1−x/ZnS core/shell 1204
quantum dots and zinc(II) phthalocyanine. J. Phys. Chem. C (2016) 120:15460-15470. doi: 1205
10.1021/acs.jpcc.6b04301 1206
160. Zhu M, Zhang J, Zhou Y, Xing P, Gong L, Su C, Qi D, Du H, Bian Y, Jiang J. Two-photon 1207
excited FRET dyads for lysosome-targeted imaging and photodynamic therapy. Inorg. 1208
Chem. (2018) 57:11537-11542. doi: 10.1021/acs.inorgchem.8b01581 1209
Jones and Bradshaw RET: Theory to Applications
41
161. Kim S, Huang H, Pudavar HE, Cui Y, Prasad PN. Intraparticle energy transfer and 1210
fluorescence photoconversion in nanoparticles: An optical highlighter nanoprobe for two-1211
photon bioimaging. Chem. Mater. (2007) 19:5650-5656. doi: 10.1021/cm071273x 1212
162. Acosta Y, Zhang Q, Rahaman A, Ouellet H, Xiao C, Sun J, Li C. Imaging cytosolic 1213
translocation of Mycobacteria with two-photon fluorescence resonance energy transfer 1214
microscopy. Biomed. Opt. Express (2014) 5:3990-4001. doi: 10.1364/BOE.5.003990 1215
163. He T, Chen R, Lim ZB, Rajwar D, Ma L, Wang Y, Gao Y, Grimsdale AC, Sun H. Efficient 1216
energy transfer under two-photon excitation in a 3D, supramolecular, Zn(II)-coordinated, 1217
self-assembled organic network. Adv. Opt. Mater. (2014) 2:40-47. doi: 1218
10.1002/adom.201300407 1219
164. Choi Y, Park Y, Kang T, Lee LP. Selective and sensitive detection of metal ions by 1220
plasmonic resonance energy transfer-based nanospectroscopy. Nat. Nanotechnol. (2009) 1221
4:742. doi: 10.1038/nnano.2009.258 1222
165. Martín-Cano D, Martín-Moreno L, García-Vidal FJ, Moreno E. Resonance energy transfer 1223
and superradiance mediated by plasmonic nanowaveguides. Nano Lett. (2010) 10:3129-1224
3134. doi: 10.1021/nl101876f 1225
166. Faessler V, Hrelescu C, Lutich AA, Osinkina L, Mayilo S, Jäckel F, Feldmann J. 1226
Accelerating fluorescence resonance energy transfer with plasmonic nanoresonators. Chem. 1227
Phys. Lett. (2011) 508:67-70. doi: 10.1016/j.cplett.2011.03.088 1228
167. Vincent R, Carminati R. Magneto-optical control of Förster energy transfer. Phys. Rev. B 1229
(2011) 83:165426. doi: 10.1103/PhysRevB.83.165426 1230
168. Cushing SK, Li J, Meng F, Senty TR, Suri S, Zhi M, Li M, Bristow AD, Wu N. 1231
Photocatalytic activity enhanced by plasmonic resonant energy transfer from metal to 1232
semiconductor. J. Am. Chem. Soc. (2012) 134:15033-15041. doi: 10.1021/ja305603t 1233
169. Gonzaga-Galeana JA, Zurita-Sánchez JR. A revisitation of the Förster energy transfer near a 1234
metallic spherical nanoparticle: (1) Efficiency enhancement or reduction? (2) The control of 1235
the Förster radius of the unbounded medium. (3) The impact of the local density of states. J. 1236
Chem. Phys. (2013) 139:244302. doi: 10.1063/1.4847875 1237
170. Schleifenbaum F, Kern AM, Konrad A, Meixner AJ. Dynamic control of Förster energy 1238
transfer in a photonic environment. Phys. Chem. Chem. Phys. (2014) 16:12812-12817. doi: 1239
10.1039/C4CP01306A 1240
Jones and Bradshaw RET: Theory to Applications
42
This is a provisional file, not the final typeset article
171. Li J, Cushing SK, Meng F, Senty TR, Bristow AD, Wu N. Plasmon-induced resonance 1241
energy transfer for solar energy conversion. Nat. Photonics (2015) 9:601. doi: 1242
10.1038/nphoton.2015.142 1243
172. Ghenuche P, Mivelle M, de Torres J, Moparthi SB, Rigneault H, Van Hulst NF, García-1244
Parajó MF, Wenger J. Matching nanoantenna field confinement to FRET distances enhances 1245
Förster energy transfer rates. Nano Lett. (2015) 15:6193-6201. doi: 1246
10.1021/acs.nanolett.5b02535 1247
173. Konrad A, Metzger M, Kern AM, Brecht M, Meixner AJ. Controlling the dynamics of 1248
Förster resonance energy transfer inside a tunable sub-wavelength Fabry–Pérot-resonator. 1249
Nanoscale (2015) 7:10204-10209. doi: 10.1039/C5NR02027A 1250
174. Tumkur TU, Kitur JK, Bonner CE, Poddubny AN, Narimanov EE, Noginov MA. Control of 1251
Förster energy transfer in the vicinity of metallic surfaces and hyperbolic metamaterials. 1252
Faraday Discuss. (2015) 178:395-412. doi: 10.1039/C4FD00184B 1253
175. Bidault S, Devilez A, Ghenuche P, Stout B, Bonod N, Wenger J. Competition between 1254
Förster resonance energy transfer and donor photodynamics in plasmonic dimer 1255
nanoantennas. ACS Photonics (2016) 3:895-903. doi: 10.1021/acsphotonics.6b00148 1256
176. de Torres J, Ferrand P, Colas des Francs G, Wenger J. Coupling emitters and silver 1257
nanowires to achieve long-range plasmon-mediated fluorescence energy transfer. ACS Nano 1258
(2016) 10:3968-3976. doi: 10.1021/acsnano.6b00287 1259
177. Poudel A, Chen X, Ratner MA. Enhancement of resonant energy transfer due to an 1260
evanescent wave from the metal. J. Phys. Chem. Lett. (2016) 7:955-960. doi: 1261
10.1021/acs.jpclett.6b00119 1262
178. Marocico CA, Zhang X, Bradley AL. A theoretical investigation of the influence of gold 1263
nanosphere size on the decay and energy transfer rates and efficiencies of quantum emitters. 1264
J. Chem. Phys. (2016) 144:024108. doi: 10.1063/1.4939206 1265
179. Wubs M, Vos WL. Förster resonance energy transfer rate in any dielectric nanophotonic 1266
medium with weak dispersion. New J. Phys. (2016) 18:053037. doi: 10.1088/1367-1267
2630/18/5/053037 1268
180. Higgins LJ, Marocico CA, Karanikolas VD, Bell AP, Gough JJ, Murphy GP, Parbrook PJ, 1269
Bradley AL. Influence of plasmonic array geometry on energy transfer from a quantum well 1270
to a quantum dot layer. Nanoscale (2016) 8:18170-18179. doi: 10.1039/C6NR05990B 1271
Jones and Bradshaw RET: Theory to Applications
43
181. Bujak Ł, Ishii T, Sharma DK, Hirata S, Vacha M. Selective turn-on and modulation of 1272
resonant energy transfer in single plasmonic hybrid nanostructures. Nanoscale (2017) 1273
9:1511-1519. doi: 10.1039/C6NR08740J 1274
182. Murphy GP, Gough JJ, Higgins LJ, Karanikolas VD, Wilson KM, Garcia Coindreau JA, 1275
Zubialevich VZ, Parbrook PJ, Bradley AL. Ag colloids and arrays for plasmonic non-1276
radiative energy transfer from quantum dots to a quantum well. Nanotechnology (2017) 1277
28:115401. doi: 10.1088/1361-6528/aa5b67 1278
183. Steele JM, Ramnarace CM, Farner WR. Controlling FRET enhancement using plasmon 1279
modes on gold nanogratings. J. Phys. Chem. C (2017) 121:22353-22360. doi: 1280
10.1021/acs.jpcc.7b07317 1281
184. Akulov K, Bochman D, Golombek A, Schwartz T. Long-distance resonant energy transfer 1282
mediated by hybrid plasmonic-photonic modes. J. Phys. Chem. C (2018) 122:15853-15860. 1283
doi: 10.1021/acs.jpcc.8b03030 1284
185. Asgar H, Jacob L, Hoang TB. Fast spontaneous emission and high Förster resonance energy 1285
transfer rate in hybrid organic/inorganic plasmonic nanostructures. J. Appl. Phys. (2018) 1286
124:103105. doi: 10.1063/1.5052350 1287
186. Eldabagh N, Micek M, DePrince AE, Foley JJ. Resonance energy transfer mediated by 1288
metal-dielectric composite nanostructures. J. Phys. Chem. C (2018) 122:18256-18265. doi: 1289
10.1021/acs.jpcc.8b04419 1290
187. Glaeske M, Juergensen S, Gabrielli L, Menna E, Mancin F, Gatti T, Setaro A. 1291
PhysicaPlasmon-assisted energy transfer in hybrid nanosystems. Phys. Status Solidi Rapid 1292
Res. Lett. (2018) 12:1800508. doi: 10.1002/pssr.201800508 1293
188. Hu J, Wu M, Jiang L, Zhong Z, Zhou Z, Rujiralai T, Ma J. Combining gold nanoparticle 1294
antennas with single-molecule fluorescence resonance energy transfer (smFRET) to study 1295
DNA hairpin dynamics. Nanoscale (2018) 10:6611-6619. doi: 10.1039/C7NR08397A 1296
189. Roth DJ, Nasir ME, Ginzburg P, Wang P, Le Marois A, Suhling K, Richards D, Zayats AV. 1297
Förster resonance energy transfer inside hyperbolic metamaterials. ACS Photonics (2018) 1298
5:4594-4603. doi: 10.1021/acsphotonics.8b01083 1299
190. Wu J-S, Lin Y-C, Sheu Y-L, Hsu L-Y. Characteristic distance of resonance energy transfer 1300
coupled with surface plasmon polaritons. J. Phys. Chem. Lett. (2018) 9:7032-7039. doi: 1301
10.1021/acs.jpclett.8b03429 1302
Jones and Bradshaw RET: Theory to Applications
44
This is a provisional file, not the final typeset article
191. Zurita-Sánchez JR, Méndez-Villanueva J. Förster energy transfer in the vicinity of two 1303
metallic nanospheres (dimer). Plasmonics (2018) 13:873-883. doi: 10.1007/s11468-017-1304
0583-4 1305
192. Olivo J, Zapata-Rodríguez CJ, Cuevas M. Spatial modulation of the electromagnetic energy 1306
transfer by excitation of graphene waveguide surface plasmons. J. Opt. (2019) 21:045002. 1307
doi: 10.1088/2040-8986/ab0ab9 1308
193. Bohlen J, Cuartero-González Á, Pibiri E, Ruhlandt D, Fernández-Domínguez AI, Tinnefeld 1309
P, Acuna GP. Plasmon-assisted Förster resonance energy transfer at the single-molecule 1310
level in the moderate quenching regime. Nanoscale (2019) 11:7674-7681. doi: 1311
10.1039/C9NR01204D 1312
194. Wang Y, Li H, Zhu W, He F, Huang Y, Chong R, Kou D, Zhang W, Meng X, Fang X. 1313
Plasmon-mediated nonradiative energy transfer from a conjugated polymer to a plane of 1314
graphene-nanodot-supported silver nanoparticles: an insight into characteristic distance. 1315
Nanoscale (2019) 11:6737-6746. doi: 10.1039/C8NR09576K 1316
195. Cortes CL, Jacob Z. Fundamental figures of merit for engineering Förster resonance energy 1317
transfer. Opt. Express (2018) 26:19371-19387. doi: 10.1364/OE.26.019371 1318
196. Pustovit VN, Shahbazyan TV. Resonance energy transfer near metal nanostructures 1319
mediated by surface plasmons. Phys. Rev. B (2011) 83:085427. doi: 1320
10.1103/PhysRevB.83.085427 1321
197. Ding W, Hsu L-Y, Schatz GC. Plasmon-coupled resonance energy transfer: A real-time 1322
electrodynamics approach. J. Chem. Phys. (2017) 146:064109. doi: 10.1063/1.4975815 1323
198. Ghenuche P, de Torres J, Moparthi SB, Grigoriev V, Wenger J. Nanophotonic enhancement 1324
of the Förster resonance energy-transfer rate with single nanoapertures. Nano Lett. (2014) 1325
14:4707-4714. doi: 10.1021/nl5018145 1326
199. Zhao L, Ming T, Shao L, Chen H, Wang J. Plasmon-controlled Förster resonance energy 1327
transfer. J. Phys. Chem. C (2012) 116:8287-8296. doi: 10.1021/jp300916a 1328
200. Hsu L-Y, Ding W, Schatz GC. Plasmon-coupled resonance energy transfer. J. Phys. Chem. 1329
Lett. (2017) 8:2357-2367. doi: 10.1021/acs.jpclett.7b00526 1330
201. Ding W, Hsu L-Y, Heaps CW, Schatz GC. Plasmon-coupled resonance energy transfer II: 1331
Exploring the peaks and dips in the electromagnetic coupling factor. J. Phys. Chem. C 1332
(2018) 122:22650-22659. doi: 10.1021/acs.jpcc.8b07210 1333
202. de Torres J, Mivelle M, Moparthi SB, Rigneault H, Van Hulst NF, García-Parajó MF, 1334
Margeat E, Wenger J. Plasmonic nanoantennas enable forbidden Förster dipole-dipole 1335
Jones and Bradshaw RET: Theory to Applications
45
energy transfer and enhance the FRET efficiency. Nano Lett. (2016) 16:6222-6230. doi: 1336
10.1021/acs.nanolett.6b02470 1337
203. Breshike CJ, Riskowski RA, Strouse GF. Leaving Förster resonance energy transfer behind: 1338
Nanometal surface energy transfer predicts the size-enhanced energy coupling between a 1339
metal nanoparticle and an emitting dipole. J. Phys. Chem. C (2013) 117:23942-23949. doi: 1340
10.1021/jp407259r 1341
204. Zhang X, Marocico CA, Lunz M, Gerard VA, Gun’ko YK, Lesnyak V, Gaponik N, Susha 1342
AS, Rogach AL, Bradley AL. Experimental and theoretical investigation of the distance 1343
dependence of localized surface plasmon coupled Förster resonance energy transfer. ACS 1344
Nano (2014) 8:1273-1283. doi: 10.1021/nn406530m 1345
205. Panman MR, Shaw DJ, Ensing B, Woutersen S. Local orientational order in liquids revealed 1346
by resonant vibrational energy transfer. Phys. Rev. Lett. (2014) 113:207801. doi: 1347
10.1103/PhysRevLett.113.207801 1348
206. Chen H, Wen X, Li J, Zheng J. Molecular distances determined with resonant vibrational 1349
energy transfers. J. Phys. Chem. A (2014) 118:2463-2469. doi: 10.1021/jp500586h 1350
207. Chen H, Bian H, Li J, Wen X, Zhang Q, Zhuang W, Zheng J. Vibrational energy transfer: 1351
An angstrom molecular ruler in studies of ion pairing and clustering in aqueous solutions. J. 1352
Phys. Chem. B (2015) 119:4333-4349. doi: 10.1021/jp512320a 1353
208. Piatkowski L, Eisenthal KB, Bakker HJ. Ultrafast intermolecular energy transfer in heavy 1354
water. Phys. Chem. Chem. Phys. (2009) 11:9033-9038. doi: 10.1039/B908975F 1355
209. Zhang Z, Piatkowski L, Bakker HJ, Bonn M. Ultrafast vibrational energy transfer at the 1356
water/air interface revealed by two-dimensional surface vibrational spectroscopy. Nat. 1357
Chem. (2011) 3:888. doi: 10.1038/nchem.1158 1358
210. Piatkowski L, de Heij J, Bakker HJ. Probing the distribution of water molecules hydrating 1359
lipid membranes with ultrafast Förster vibrational energy transfer. J. Phys. Chem. B (2013) 1360
117:1367-1377. doi: 10.1021/jp310602v 1361
211. Yang M, Li F, Skinner JL. Vibrational energy transfer and anisotropy decay in liquid water: 1362
Is the Förster model valid? J. Chem. Phys. (2011) 135:164505. doi: 10.1063/1.3655894 1363
212. Rustomji K, Dubois M, Kuhlmey B, de Sterke CM, Enoch S, Abdeddaim R, Wenger J. 1364
Direct imaging of the energy-transfer enhancement between two dipoles in a photonic 1365
cavity. Phys. Rev. X (2019) 9:011041. doi: 10.1103/PhysRevX.9.011041 1366
213. Cederbaum LS, Zobeley J, Tarantelli F. Giant intermolecular decay and fragmentation of 1367
clusters. Phys. Rev. Lett. (1997) 79:4778-4781. doi: 10.1103/PhysRevLett.79.4778 1368
Jones and Bradshaw RET: Theory to Applications
46
This is a provisional file, not the final typeset article
214. Marburger S, Kugeler O, Hergenhahn U, Möller T. Experimental evidence for interatomic 1369
coulombic decay in Ne clusters. Phys. Rev. Lett. (2003) 90:203401. doi: 1370
10.1103/PhysRevLett.90.203401 1371
215. Santra R, Zobeley J, Cederbaum LS, Moiseyev N. Interatomic coulombic decay in van der 1372
Waals clusters and impact of nuclear motion. Phys. Rev. Lett. (2000) 85:4490-4493. doi: 1373
10.1103/PhysRevLett.85.4490 1374
216. Scheit S, Cederbaum LS, Meyer HD. Time-dependent interplay between electron emission 1375
and fragmentation in the interatomic Coulombic decay. J. Chem. Phys. (2003) 118:2092-1376
2107. doi: 10.1063/1.1531996 1377
217. Jahnke T. Interatomic and intermolecular Coulombic decay: the coming of age story. J. 1378
Phys. B: At. Mol. Opt. Phys. (2015) 48:082001. doi: 10.1088/0953-4075/48/8/082001 1379
218. Fasshauer E. Non-nearest neighbour ICD in clusters. New J. Phys. (2016) 18:043028. doi: 1380
10.1088/1367-2630/18/4/043028 1381
219. Dolbundalchok P, Peláez D, Aziz EF, Bande A. Geometrical control of the interatomic 1382
coulombic decay process in quantum dots for infrared photodetectors. J. Comput. Chem. 1383
(2016) 37:2249-2259. doi: 10.1002/jcc.24410 1384
220. Haller A, Chiang Y-C, Menger M, Aziz EF, Bande A. Strong field control of the interatomic 1385
Coulombic decay process in quantum dots. Chem. Phys. (2017) 482:135-145. doi: 1386
10.1016/j.chemphys.2016.09.020 1387
221. Goldzak T, Gantz L, Gilary I, Bahir G, Moiseyev N. Vertical currents due to interatomic 1388
Coulombic decay in experiments with two coupled quantum wells. Phys. Rev. B (2016) 1389
93:045310. doi: 10.1103/PhysRevB.93.045310 1390
222. Dreuw A, Faraji S. A quantum chemical perspective on (6-4) photolesion repair by 1391
photolyases. Phys. Chem. Chem. Phys. (2013) 15:19957-19969. doi: 10.1039/C3CP53313A 1392
223. Harbach PHP, Schneider M, Faraji S, Dreuw A. Intermolecular coulombic decay in biology: 1393
The initial electron detachment from FADH– in DNA photolyases. J. Phys. Chem. Lett. 1394
(2013) 4:943-949. doi: 10.1021/jz400104h 1395
224. Hemmerich JL, Bennett R, Buhmann SY. The influence of retardation and dielectric 1396
environments on interatomic Coulombic decay. Nat. Commun. (2018) 9:2934. doi: 1397
10.1038/s41467-018-05091-x 1398
225. Bennett R, Votavová P, Kolorenč P, Miteva T, Sisourat N, Buhmann SY. Virtual photon 1399
approximation for three-body interatomic coulombic decay. Phys. Rev. Lett. (2019) 1400
122:153401. doi: 10.1103/PhysRevLett.122.153401 1401
Jones and Bradshaw RET: Theory to Applications
47
226. Beljonne D, Curutchet C, Scholes GD, Silbey RJ. Beyond Förster resonance energy transfer 1402
in biological and nanoscale systems. J. Phys. Chem. B (2009) 113:6583-6599. doi: 1403
10.1021/jp900708f 1404
227. Xia Z, Rao J. Biosensing and imaging based on bioluminescence resonance energy transfer. 1405
Curr. Opin. Biotechnol. (2009) 20:37-44. doi: 10.1016/j.copbio.2009.01.001 1406
228. Ling J, Huang CZ. Energy transfer with gold nanoparticles for analytical applications in the 1407
fields of biochemical and pharmaceutical sciences. Anal. Methods (2010) 2:1439-1447. doi: 1408
10.1039/C0AY00452A 1409
229. Scholes GD, Fleming GR, Olaya-Castro A, van Grondelle R. Lessons from nature about 1410
solar light harvesting. Nat. Chem. (2011) 3:763-774. doi: 10.1038/nchem.1145 1411
230. Şener M, Strümpfer J, Hsin J, Chandler D, Scheuring S, Hunter CN, Schulten K. Förster 1412
energy transfer theory as reflected in the structures of photosynthetic light-harvesting 1413
systems. ChemPhysChem (2011) 12:518-531. doi: 10.1002/cphc.201000944 1414
231. Chen N-T, Cheng S-H, Liu C-P, Souris JS, Chen C-T, Mou C-Y, Lo L-W. Recent advances 1415
in nanoparticle-based Förster resonance energy transfer for biosensing, molecular imaging 1416
and drug release profiling. Int. J. Mol. Sci. (2012) 13:16598-16623. 1417
232. Lohse MJ, Nuber S, Hoffmann C. Fluorescence/bioluminescence resonance energy transfer 1418
techniques to study G-protein-coupled receptor activation and signaling. Pharmacol. Rev. 1419
(2012) 64:299-336. doi: 10.1124/pr.110.004309 1420
233. Yang J, Yoon M-C, Yoo H, Kim P, Kim D. Excitation energy transfer in multiporphyrin 1421
arrays with cyclic architectures: towards artificial light-harvesting antenna complexes. 1422
Chem. Soc. Rev. (2012) 41:4808-4826. doi: 10.1039/C2CS35022J 1423
234. Zadran S, Standley S, Wong K, Otiniano E, Amighi A, Baudry M. Fluorescence resonance 1424
energy transfer (FRET)-based biosensors: visualizing cellular dynamics and bioenergetics. 1425
Appl. Microbiol. Biotechnol. (2012) 96:895-902. doi: 10.1007/s00253-012-4449-6 1426
235. Fassioli F, Dinshaw R, Arpin PC, Scholes GD. Photosynthetic light harvesting: excitons and 1427
coherence. J. R. Soc. Interface (2014) 11:20130901. doi: doi:10.1098/rsif.2013.0901 1428
236. Chenu A, Scholes GD. Coherence in energy transfer and photosynthesis. Annu. Rev. Phys. 1429
Chem. (2015) 66:69-96. doi: 10.1146/annurev-physchem-040214-121713 1430
237. Peng H-Q, Niu L-Y, Chen Y-Z, Wu L-Z, Tung C-H, Yang Q-Z. Biological applications of 1431
supramolecular assemblies designed for excitation energy transfer. Chem. Rev. (2015) 1432
115:7502-7542. doi: 10.1021/cr5007057 1433
Jones and Bradshaw RET: Theory to Applications
48
This is a provisional file, not the final typeset article
238. Brédas J-L, Sargent EH, Scholes GD. Photovoltaic concepts inspired by coherence effects in 1434
photosynthetic systems. Nat. Mater. (2016) 16:35-44. doi: 10.1038/nmat4767 1435
239. Jiang Y, McNeill J. Light-harvesting and amplified energy transfer in conjugated polymer 1436
nanoparticles. Chem. Rev. (2017) 117:838-859. doi: 10.1021/acs.chemrev.6b00419 1437
240. Mirkovic T, Ostroumov EE, Anna JM, van Grondelle R, Govindjee, Scholes GD. Light 1438
absorption and energy transfer in the antenna complexes of photosynthetic organisms. 1439
Chem. Rev. (2017) 117:249-293. doi: 10.1021/acs.chemrev.6b00002 1440
241. Scholes GD, Fleming GR, Chen LX, Aspuru-Guzik A, Buchleitner A, Coker DF, Engel GS, 1441
van Grondelle R, Ishizaki A, Jonas DM, Lundeen JS, McCusker JK, Mukamel S, Ogilvie JP, 1442
Olaya-Castro A, Ratner MA, Spano FC, Whaley KB, Zhu X. Using coherence to enhance 1443
function in chemical and biophysical systems. Nature (2017) 543:647. doi: 1444
10.1038/nature21425 1445
242. Su Q, Feng W, Yang D, Li F. Resonance energy transfer in upconversion nanoplatforms for 1446
selective biodetection. Acc. Chem. Res. (2017) 50:32-40. doi: 10.1021/acs.accounts.6b00382 1447
243. Jumper CC, Rafiq S, Wang S, Scholes GD. From coherent to vibronic light harvesting in 1448
photosynthesis. Curr. Opin. Chem. Biol. (2018) 47:39-46. doi: 10.1016/j.cbpa.2018.07.023 1449
244. Lerner E, Cordes T, Ingargiola A, Alhadid Y, Chung S, Michalet X, Weiss S. Toward 1450
dynamic structural biology: Two decades of single-molecule Förster resonance energy 1451
transfer. Science (2018) 359:eaan1133. doi: 10.1126/science.aan1133 1452
245. El Khamlichi C, Reverchon-Assadi F, Hervouet-Coste N, Blot L, Reiter E, Morisset-Lopez 1453
S. Bioluminescence resonance energy transfer as a method to study protein-protein 1454
interactions: Application to G-protein coupled receptor biology. Molecules (2019) 24:537. 1455
246. Cupellini L, Corbella M, Mennucci B, Curutchet C. Electronic energy transfer in 1456
biomacromolecules. WIREs Comput Mol Sci. (2019) 9:e1392. doi: 10.1002/wcms.1392 1457
247. Power EA, Thirunamachandran T. Quantum electrodynamics in a cavity. Phys. Rev. A 1458
(1982) 25:2473-2484. doi: 10.1103/PhysRevA.25.2473 1459
248. Mabuchi H, Doherty AC. Cavity quantum electrodynamics: Coherence in context. Science 1460
(2002) 298:1372-1377. doi: 10.1126/science.1078446 1461
249. Miller R, Northup TE, Birnbaum KM, Boca A, Boozer AD, Kimble HJ. Trapped atoms in 1462
cavity QED: coupling quantized light and matter. J. Phys. B: At. Mol. Opt. Phys. (2005) 1463
38:S551-S565. doi: 10.1088/0953-4075/38/9/007 1464
250. Walther H, Varcoe BTH, Englert B-G, Becker T. Cavity quantum electrodynamics. Rep. 1465
Prog. Phys. (2006) 69:1325-1382. doi: 10.1088/0034-4885/69/5/r02 1466
Jones and Bradshaw RET: Theory to Applications
49
251. Hohenester U. Cavity quantum electrodynamics with semiconductor quantum dots: Role of 1467
phonon-assisted cavity feeding. Phys. Rev. B (2010) 81:155303. doi: 1468
10.1103/PhysRevB.81.155303 1469
252. Chang DE, Jiang L, Gorshkov AV, Kimble HJ. Cavity QED with atomic mirrors. New J. 1470
Phys. (2012) 14:063003. doi: 10.1088/1367-2630/14/6/063003 1471
253. Andrew P, Barnes WL. Förster energy transfer in an optical microcavity. Science (2000) 1472
290:785-788. doi: 10.1126/science.290.5492.785 1473
254. Hutchison JA, Schwartz T, Genet C, Devaux E, Ebbesen TW. Modifying chemical 1474
landscapes by coupling to vacuum fields. Angew. Chem. Int. Ed. (2012) 51:1592-1596. doi: 1475
10.1002/anie.201107033 1476
255. Thomas A, George J, Shalabney A, Dryzhakov M, Varma SJ, Moran J, Chervy T, Zhong X, 1477
Devaux E, Genet C, Hutchison JA, Ebbesen TW. Ground-state chemical reactivity under 1478
vibrational coupling to the vacuum electromagnetic field. Angew. Chem. Int. Ed. (2016) 1479
55:11462-11466. doi: 10.1002/anie.201605504 1480
256. Feist J, Garcia-Vidal FJ. Extraordinary exciton conductance induced by strong coupling. 1481
Phys. Rev. Lett. (2015) 114:196402. doi: 10.1103/PhysRevLett.114.196402 1482
257. Schachenmayer J, Genes C, Tignone E, Pupillo G. Cavity-enhanced transport of excitons. 1483
Phys. Rev. Lett. (2015) 114:196403. doi: 10.1103/PhysRevLett.114.196403 1484
258. Zhong X, Chervy T, Wang S, George J, Thomas A, Hutchison JA, Devaux E, Genet C, 1485
Ebbesen TW. Non-radiative energy transfer mediated by hybrid light-matter states. Angew. 1486
Chem. Int. Ed. (2016) 55:6202-6206. doi: 10.1002/anie.201600428 1487
259. Zhong X, Chervy T, Zhang L, Thomas A, George J, Genet C, Hutchison JA, Ebbesen TW. 1488
Energy transfer between spatially separated entangled molecules. Angew. Chem. Int. Ed. 1489
(2017) 56:9034-9038. doi: 10.1002/anie.201703539 1490
260. Georgiou K, Michetti P, Gai L, Cavazzini M, Shen Z, Lidzey DG. Control over energy 1491
transfer between fluorescent BODIPY dyes in a strongly coupled microcavity. ACS 1492
Photonics (2018) 5:258-266. doi: 10.1021/acsphotonics.7b01002 1493
261. Du M, Martínez-Martínez LA, Ribeiro RF, Hu Z, Menon VM, Yuen-Zhou J. Theory for 1494
polariton-assisted remote energy transfer. Chem. Sci. (2018) 9:6659-6669. doi: 1495
10.1039/C8SC00171E 1496
262. Schäfer C, Ruggenthaler M, Appel H, Rubio A. Modification of excitation and charge 1497
transfer in cavity quantum-electrodynamical chemistry. Proc. Natl. Acad. Sci. USA (2019) 1498
116:4883-4892. doi: 10.1073/pnas.1814178116 1499
Jones and Bradshaw RET: Theory to Applications
50
This is a provisional file, not the final typeset article
263. Hertzog M, Wang M, Mony J, Börjesson K. Strong light–matter interactions: a new 1500
direction within chemistry. Chem. Soc. Rev. (2019) 48:937-961. doi: 10.1039/C8CS00193F 1501
264. Juzeliūnas. G, Andrews DL. Quantum electrodynamics of resonant energy-transfer in 1502
condensed matter. Phys. Rev. B (1994) 49:8751-8763. doi: 10.1103/PhysRevB.49.8751 1503
265. Juzeliūnas. G, Andrews DL. Quantum electrodynamics of resonant energy-transfer in 1504
condensed matter. II. dynamical aspects. Phys. Rev. B (1994) 50:13371-13378. doi: 1505
10.1103/PhysRevB.50.13371 1506
266. Dung HT, Knöll L, Welsch D-G. Intermolecular energy transfer in the presence of 1507
dispersing and absorbing media. Phys. Rev. A (2002) 65:043813. doi: 1508
10.1103/PhysRevA.65.043813 1509
267. Cortes CL, Jacob Z. Super-Coulombic atom–atom interactions in hyperbolic media. Nat. 1510
Commun. (2017) 8:14144. doi: 10.1038/ncomms14144 1511
268. Newman WD, Cortes CL, Afshar A, Cadien K, Meldrum A, Fedosejevs R, Jacob Z. 1512
Observation of long-range dipole-dipole interactions in hyperbolic metamaterials. Sci. Adv. 1513
(2018) 4:eaar5278. doi: 10.1126/sciadv.aar5278 1514
269. Mahmoud AM, Liberal I, Engheta N. Dipole-dipole interactions mediated by epsilon-and-1515
mu-near-zero waveguide supercoupling. Opt. Mater. Express (2017) 7:415-424. doi: 1516
10.1364/OME.7.000415 1517
270. Scheel S, Buhmann S. Macroscopic quantum electrodynamics-concepts and applications. 1518
Acta Phys. Slov. (2008) 58:675-809. doi: 10.2478/v10155-010-0092-x 1519
1520
Jones and Bradshaw RET: Theory to Applications
51
Table 1. All the system states and their associated energies for RET. The energies of the donor 1521
and acceptor are represented by superscript of DE and AE , respectively. Due to conservation 1522
of energy arguments, =n mE E . 1523
System state Dirac bracket Energy
i ( )0, ;0 ,n
D AE E p λ� 0+n
D AE E
1I ( )0 0, ;1 ,D AE E p λ� 0 0
D AE E cp+ + ℏ
2I ( ), ;1 ,n m
D AE E p λ�
n m
D AE E cp+ + ℏ
f ( )0 , ;0 ,m
D AE E p λ�
0 + m
D AE E
1524
Figure 1. Representation of energy transfer, the excited donor (on the left-hand side) transfers 1525
energy, represented by the red arrow, to the acceptor (on the right). 1526
Figure 2. Two time-orderings for RET between a donor (D) and an acceptor (A). The vertical 1527
lines denote the two molecules, wavy lines are the photons, n and m represents the excited state 1528
of D and A, respectively, and 0 is their ground state; time, t, increases up the graph. Red, black 1529
and blue lines represent the initial, intermediate and final system state. 1530
Figure 3. One of 24 possible time-orderings for RET mediated by a third molecule, M, acting 1531
as a bridge between donor D and acceptor A. Energy is transferred from D to A, and M begins 1532
and ends in its ground state. 1533
Figure 4. Two-step RET in a second-generation phenylacetylene dendrimer. This schematic 1534
depicts initial electronic excitation at a peripheral phenyl group, which acts as a donor of 1535
energy to a neighbouring inner-ring chromophore; this acceptor then becomes a donor of 1536
energy to the phenaline core. Original image appeared in reference [137]. 1537
Jones and Bradshaw RET: Theory to Applications
52
This is a provisional file, not the final typeset article
Figure 5. Representation of energy pooling, the two excited donors (on the left- and right-hand 1538
side) transfer energy, represented by the red arrows, to the acceptor (in the centre). 1539
Figure 6. (a) Photoionization of a neon dimer, via ejection of an inner shell electron from an 1540
atom (green arrow), due to incident x-ray radiation (orange wavy line). (b) Interatomic 1541
Coulombic decay: an outer electron relaxes into the vacancy (blue arrow) and, consequently, 1542
photo-ionization of the other atom occurs due to energy transfer between the atoms (red arrow). 1543
(c) The newly charged atoms (plus signs) repel each other (yellow arrows), which results in 1544
destruction of the neon dimer. 1545
1546
Jones and Bradshaw RET: Theory to Applications
53
1547
1548
Figure 1. 1549
1550
1551
Figure 2. 1552
1553
Jones and Bradshaw RET: Theory to Applications
54
This is a provisional file, not the final typeset article
1554
1555
Figure 3. 1556
1557
1558
1559
Figure 4. 1560
1561
Jones and Bradshaw RET: Theory to Applications
55
1562
1563
Figure 5. 1564
1565
1566
1567
Figure 6. 1568
1569